I think this is a very complex question because when we talk about buddhism and the west, we first have to look at the origins of buddhism, its spread, and changes over the centuries after its general founding. This is difficult because buddhism in southeast Asia is different then buddhism in China than buddhism in Korea than buddhism in Japan, Tibet, etc. Then you look at Theravada vs Mahayana vs Vajrayana and the myriad of sects that exist, each adding or removing or interpreting things in a different way. I'm not an expert on any of these, but have looked at the history of buddhism in Japan, especially, due to studies in Japanese and martial arts history. I will try to answer this along those lines.

Asoka Maurya in the 3rd century BC played a major role in the spread of buddhism throughout India. Recall that his grandfather, Candragupta, was the founder of the Maurya Empire which saw consolidation and unification of the shattered Indian states by Alexander the Great roughly five years earlier. His son, Bindusara, expanded the empire further, and then it was Asoka's conversion that sealed the deal. It's said that Asoka supported tens of thousands of monks and even more stupas on top of countless monasteries. And when the teachings were brought to Ceylon, that early buddhist canon was committed to writing in the Ceylonese Mahavamsa records, where it was written-

the most learned monks handed down the text and commentary of the Three Pitakas (Tipitaka) orally; but since they perceived that the people were falling away from the orthodox teaching, the monks convened; and so that the true doctrines might endure, they wrote them down in books

The result of this is the Pali canon. Therein lies the base for Theravada buddhism. Included in these doctrines are rules and guidance for daily affairs with the sangha, thoughts on how to maintain communal harmony within a large and spiritually diverse community, and more teachings on how to actualize sunyata than there are grains of sand on the beach. And throughout it all, the goal was one of "happiness", for that pure, true happiness brought about by realization of sunyata awakens one into the next world (so to speak)-

One should learn virtue which is of extensive goal, and which hat the faculty of Happiness; and one should devote oneself to Charity, to tranquil behavior and to thoughts of Friendship. Having devoted himself to these three virtues, which provide reason for happiness, a wise man gaineth the world of happiness- a world all free from distress.

Secularity of these teachings was crucial to the spread through India and abroad. Asoka sent teachers of the buddhist dharma to the West: Antiochus II of Syria, Ptolemy II of Egypt, Magas of Cyrene, Antigonus Gonatas of Macedonia, and Alexander II of Epirus. This ties in to another question I answered awhile ago on why didn't Hinduism, Buddhism, Taoism, or any religion from East Asia spread to the Europe Continent or Middle East via the Silk Road or any trade routes. As J. Kennedy wrote in the Journal of the Royal Asiatic Society of Great Britain and Ireland in 1902, Asoka's emissaries are marked in his edicts, but there appear to be no real records on the Western side of such transmissions- or rather, those transmissions taking hold. This has the effect of keeping details of buddhist philosophy consolidated to Asia, out of the field of awareness of Western scholars even if the Rock Edicts of Asoka were available from the Indo-Greek King Menander I.

The first Christian notice of Buddhism that I'm aware of is with Clement of Alexandria in 212AD. In his Stromata (ie, Miscellanies), he recognizes some Indians who obey the precepts of Boutta, "whom on the account of his extraordinary sanctity they have raised to divine honours." Clement spoke highly of Buddhism, calling it a philosophy "first in its ranks" for shedding light over the nations." It's important to understand how buddhism didn't really flow past the Middle East in the pre-medival times. With the Silk Road example, the Turkharistan and Sogdiana regions in the Mid East acted as major trading hubs on the Silk Road between the four directions. All cultures religions and philosophies aggregated there in the 1st century AD. Diversity existed in part because the various city-states of the region were not part of a single polity nor did they tend to have complete control over their subjects. Thus no one single religion/philosophy/politic/etc could exert dominance. It should also be noted that this lack of cohesion resulted in differing rituals even among practitioners of the same religion; Zoroastrians and Buddhists in Sogdian cities practiced differently than those in Iran, for example. Locality plays a major role in evolution of religion, philosophy, and other cultural aspects after all.

Why does all this matter? Because of your question regarding the West and the idea of religion vs philosophy. When the West thinks of religion, they think of divinity, something greater to worship, with teachings and ritual to bring one closer to that divinity. When Jacinto Orfanel, a Spanish Dominican, arrived in Japan in 1607 to preach Christianity, he (along with other missionaries to Japan) reported back how the Japanese were-

much addicted to their idolatries" and "there are some people who worship and idol called Shaka [Sakyamuni Buddha]... other people worship an idol called Amida [Amida Buddha].

Herein lies the crux to the confusion with the Western view of buddhism as religion vs philosophy. This "worship" is not really worship. Amida is not worshipped as an external deity no more than Siddhartha or Sakyamuni. Amida, Dainichi, Kokuzo, all the Vidyaraja, are aspects of the self, with "worship" actually being a turn to focus on those aspects, or manifesting as that right here and now. This secular, philosophical nature is what allowed it to spread to Daoist China and Shinto/Onmyodo-based Japan: association of similarities and acceptance of what's already there.... with a different twist. (continued below)

Finally, a question I can answer! The short answer is that Buddhism, especially historical Buddhism, was more of a philosophy and less of a religion than Christianity. That’s not to say that it’s not a religion – just that many sects of Buddhism believe in more questioning and philosophical musing than most sects of Christianity. Westerners like Alan Watts have relatively recently (early to mid 20th century) revived the historical tradition of blending Buddhism with other religions and philosophies.

Now, the long answer. When Siddhartha Buddha conceived Buddhism in around 520 BCE, he laid it out primarily on two rule sets. The first was the Four Noble Truths, and the second the Eightfold Path. I’ll begin by explaining each of these 12 essential elements of Buddhism, as I think you’ll see the philosophical leanings. I’ll pull these 12 elements from Buddhist Scriptures, edited by Donald Lopez, but they are widely available.

Now this, followers, is the noble truth of suffering: birth is suffering, aging is suffering, illness is suffering, death is suffering; union with what is displeasing is suffering; separation from what is pleasing is suffering; not to get what one wants is suffering; in brief, the five aggregates subject to clinging are suffering.

Dukkha, the first Noble Truth, holds that all of life involves suffering. No matter how wonderful life is at any given moment, it will return to suffering at some point.

Now this, followers, is the noble truth of the origin of suffering: it is this craving which leads to re-becoming, accompanied by delight and lust, seeking delight here and there; that is, craving for sensual pleasures, craving for becoming, craving for disbecoming.

Samudaya, the second Noble Truth, states that the reason for suffering is craving and desire. Because we want and cannot have, we suffer.

Now this, followers, is the noble truth of the cessation of suffering: it is the remainderless fading away and cessation of that same craving, the giving up and relinquishing of it, freedom from it, non-reliance on it.

Nirodha, the third Noble Truth, states that there is a bit of a loophole – suffering evaporated when you rid yourself of desire. Eliminate craving and suffering follows.

Now this, followers, is the noble truth of the way leading to the cessation of suffering: it is this noble eightfold path; that is, right view, right intention, right speech, right action, right livelihood, right effort, right mindfulness, right concentration.

Marga, the fourth Noble Truth, is mostly just a plug for the Eightfold Path. In other words, there is a laid-out way to eliminate craving, and it’s called the Eightfold Path.

Already you should be able to see some major differences between Buddhism and Christianity. Nowhere in the Noble Truths are there mystical elements or strict rules laid out by a deity. While many sects do emphasize these elements, it says a lot that the fundamental basis for Buddhism, the Four Noble Truths, are essentially musings by one person who figured this stuff out. Anyway, on to the rules he laid out, which do involve those mystical elements – the Eightfold Path.

First, Right View essentially says you should start by believing in Buddhism. Basically, that karmic retribution and reincarnation are things. These are the core mystical elements of Buddhism that people have often chosen to leave out of their secular practice.

Second, Right Intention. Legitimately feel and believe that which you practice – love, don’t just act like you love; truly be content with next to nothing. This is in stark contrast to historical Catholic beliefs that actions were supremely important and actual beliefs or feelings negligible.

Third, Right Speech. Basically, don’t swear or lie. Be kind and truthful with your words.

Fourth, Right Action. Act in a way that improves your wellbeing and others. Do not kill, steal, engage sexually, or generally have material desires.

Fifth, Right Livelihood. Do not exploit others – release your material goods and live like a pauper.

Sixth, Right Effort. Focus on reducing undesirable qualities in yourself and increasing desirable qualities. Make an effort to better yourself.

Seventh, Right Mindfulness. Stay present and active in your mind. Keep a clear head and be aware of yourself and your surroundings. This is the basis for the recent Mindfulness movement, which is another example of the philosophical elements of Buddhism being integrated into western society.

Eighth, Right Concentration. Essentially, attain oneness – usually by meditating. This is a particularly difficult point to translate as it involves incorporating the mind, body, and world into a sort of transcendent state. This concentration and oneness should permeate life beyond just periods of meditation.

One of the most important aspects of these elements of Buddhism is that the Buddha never intended them to be followed as strictly as, say, the Ten Commandments. Buddhist spiritual leaders traditionally guide followers on a path, they don’t shepherd or martyr. Much of the early elements of Buddhism are told through stories; in one such story a disciple of the Buddha tells him that he is the greatest teacher of all. The Buddha scolds his disciple for making such an assumption, as there may be many more teachers who know much more and can teach much better than the Buddha. In other words, while the Four Noble Truths and Eightfold Path together form the Buddha’s path to enlightenment, one may find their own path through a totally different method and still reach nirvana. (For more information on this, try Spiritual Authority: A Buddhist Perspective)

While the fundamental basis of Buddhism carries this degree of flexibility, it’s important to note that many modern sects function as their own unique paths to enlightenment, and therefore do not allow flexibility. This is, in my opinion, the cause of the philosophy/religion confusion surrounding Buddhism: the basis for Buddhism is more philosophical in that it encourages a variety of methods and experimentation to attain true happiness, but almost all established sects believe they have found the most efficient, effective, and wide-reaching path and so follow it with religious adherence.

I’ll take an aside here to give one historical example of Buddhism functioning more as a philosophy than a religion. When Buddhism was first introduced to Japan, it was immediately shot down as heretical and counter to the local Shinto faith. On several occasions around the 6th century AD, Buddhist monks attempted to introduce the religion to Japan, and each time they were shut down both by the people and the government. Success eventually came in an unorthodox fashion – Buddhist monks began spending significant amounts of time in Shinto shrines. They would return and claim to have converted the kami, or minor deities, to Buddhism. They would explain the conversion and the philosophy behind Buddhism, and to many people this made sense. Eventually this method resulted in the widespread establishment of Buddhism, to the point where the vast majority of Japan today practices some form of Buddhism (usually in conjunction with Shinto). In fact, it was Japan that developed the sect of Buddhism most famous in the West – Zen Buddhism. For further reading on this topic, check out Kenji Matsuo’s A History of Japanese Buddhism.

This adaptation of Buddhism is a historical tradition that has in the past century taken place in the West through the likes of Alan Watts. Just as the monks in Japan framed Buddhism as functioning alongside Shinto, so did Watts frame it as functioning alongside traditional Western values – he provided the framework for a Buddhism that could exist with Christianity, or Atheism, or Hippies, or any number of other Western religions and philosophies. If you’d like to read more on his beliefs, try his seminal work, The Way of Zen. In this quite popular book, he lays out the same sentiment you’ve heard plenty of times – mindfulness brings happiness, releasing the material results in a content existence, enlightenment is a positive experience attainable by anyone. Just as the monks in Japan did, he took Buddhism and framed it as a philosophy to Trojan Horse it into a society with a radically different set of beliefs.

Anyway, I’d be happy to answer more questions on the topic, or provide more sources if you’d like! Hopefully I covered the essentials here.

While there have been some great answers so far on the general history of Buddhism and its contact with the west, I feel most of them come from an "insider perspective", i.e. from those who see no great disconnect between the religion of Buddhism as we know it in the West today, and the Buddhist traditions of Asia, especially prior to western colonisation. OP clearly doesn't share this view, and since I agree with them, I guess it falls to me to take a stab at answering the original questions: When did the West started to see Buddhism as a philosophy of enlightenment and meditation, apparently completely misunderstood by its practitioners in Asia? And is there a patient zero, someone who is (mostly) responsible for spreading this view? I'd say yes, there is.

Henry Steel Olcott (1832 - 1907) is one of the major figures of what is sometimes known as "Protestant Buddhism", the Buddhist revival movement originating in Sri Lanka with heavy nineteenth-century Anglo-American Protestant influences. Born as a Presbyterian in Orange, New Jersey, Olcott co-founded the Theosophical Society in 1874, along with the famous Russian occultist Helena Petrovna Blavatsky. In 1880, he and Blavatsky publically converted to Buddhism, the first known American to do so. After that, in 1883, Olcott traveled to Ceylon and from there he started his life's work of establishing Buddhism as a unified religion, writing countless tracts and, among other things, established the international Buddhist flag design still used today.

Olcott's own idea of Buddhism was very much shaped by his time and background as a Theosophian and Orientalist. Buddhism, in his mind, was a "scientific religion", unlike the obscurantism of religions like Christianity. He based this view on his own interpretation of ancient Buddhist scriptures, which, in true Protestant fashion, he believed to be the "real" Buddhism, unlike the primitive traditions of his Asian contemporaries, whose practices he denounced in his Buddhist Catechism as 'postively repugnant... Pagan, mean and spurious'. To him, Buddhism was 'a pure moral philosophy, a system of ethics', where rituals, charms and incantations were but perversions of its original message, the following of the five precepts. Accordingly, he also disapproved of the monastic traditions common in many Buddhist schools of thought.

Olcott's interpretation of Buddhism greatly influenced Ceylon reformists like Anagarika Dharmapala, as well as his fellows in the Theosophical Society. Through them, it eventually spread to Japan, shaping its own Neo-Buddhism and leading to a curious feedback loop where western scholars saw their own views on Buddhism confirmed through a system that they themselves helped to change.

On that basis, I would nominate Olcott as the "patient zero" of that particular interpretation of Buddhism so endemic to the West. Though probably not the first, he was certainly one of the most influential of the early Buddhist enthusiasts.

EDIT: Fixed some typos.

Sources:

• Bechert, Heinz. Der Buddhismus in Süd- und Südostasien: Geschichte und Gegenwart. Kohlhammer, Stuttgart, 2013
• Gombrich, Richard & Obeyesekere Gananath. Buddhism transformed. Princeton University Press, Princeton, 1988
• Prothero, Stephen. Henry Steel Olcott and "Protestant Buddhism" Journal of the American Academy of Religion, 1995.

Like /u/SteveGladstone said, this is a really difficult question to answer because "Buddhism" contains under it so much diversity of thought and tradition and practice. As an undergrad who's studied some buddhist thought and buddhist history I hope I can elaborate more on some elements that the other commenters haven't covered.

Most Buddhist scholars today will talks about something called "Buddhist Modernism" that describes a phenomenon very similar to what you have mentioned. It's the conception of Buddhism that you've probably seen many times in your daily life nowadays that presents Buddhism as a religion free of superstition, compatible with science, promoting an ethics of compassion and responsibility and mindfulness, divergent from what seems like a "superstitious" and "ritualistic" traditional Buddhism that's practiced widely across Asia. Historians will usually trace the emergence of this conception back to the encounter between Buddhism, as practiced in all sorts of variegated traditions across Asia, and europeans, who arrived in the colonial period with western science, philosophy, religion. Despite the diversity of Buddhist traditions across Asia, in almost every country and every Buddhist tradition during this period, the encounter will create a "modernized" Buddhist tradition.

I don't know as much about other countries (the book "The Making of Buddhist Modernism" is really great and touches on them in much more depth than I can), but Sri Lanka in the 1800s provides a really representative example of this phenomenon. When British protestant missionaries arrived in Sri Lanka, they immediately began setting up schools and missions and printing presses, with backing from the British government. To aid in their proselytization efforts, two British Methodists in the 1830s, Daniel Gogerly, and David de Silva, who had learned both Sinhala (the vernacular Sri Lankan language) and Pali (the ancient language of Theravada Buddhist text), wrote pamplets in Sinhala attacking Buddhist doctrine, questioning the historicity of Buddhist scriptures, and explaining the scientifically impossible nature of Buddhist heavens and hells. Because these pamplets were published at a mass scale in the vernacular language, Theravada monks were compelled to respond, and when they joined the debate, they questioned Christian missionaries on very much the same grounds, arguing against the historicity of Christian scripture, asking how Christian cosmology was possible scientifically, and countering Christian accusations of superstition by calling attention to aspects of Christian dogma like the Trinity. In this debate and ones that follow it, a trend develops where Sri Lankan Buddhists argue against Protestant Christianity using the very same arguments British missionaries deployed against Buddhism, but then going one step further and arguing that it is actually Buddhism instead of Christianity that is most compatible with enlightenment ideas of science, rationality, compassion. Alongside this, there arises a tendency to jettison the 'superstitious' and folk religion/lay aspects of Sri Lankan Buddhism that didn't fit into the modernist religious conception alongside a newer focus of religion on the laity rather than monasticism.

Within this environment you have the emergence of younger Buddhist reformers born to a middle class Sri Lankan families and educated in Protestant schools, most notably among them Anagarika Dharmapala, who argued at the 1893 world fair for the rational and ethical core within Buddhism, the compatibility between Buddhism and science (in fact arguing that Buddhist ideas about causality and empiricism preceded the European scientific tradition by 2,500 years) and promoted lay meditation (previously meditation was only practiced by monks). These arguments were widely received and joined transcripts of previous debates in Sri Lanka published by the Theosophical society and other European popularizers of Modernist Buddhism, and marked the beginning of the dissemination of these modernist Buddhist ideas in America and Europe.

There's a similar process that happens all across the Buddhist world, in Thailand (where the king Mongkut, after a western education, enters monastic life at age 20 following Thai tradition, only to be appalled at what he sees as the degradation of monastic tradition. He founds a new official Buddhist sect in response that's both fundamentalist in how it goes back to the Pali canon and modernist in its compatibility with 'science' and 'reason'), in Japan (I can't remember the book anymore, but there's a really great academic text about how the Meiji Restoration reformers wanted to ditch Buddhism altogether because they thought it was incompatible with their modernizing project, but the educated proponents of one modernized school of zen convinced them otherwise), and other similar cases in China and Tibet. Of course, this initial phenomenon of modernizing figures is only the beginning in a longer history of Buddhist modernism that continues to today, but I've got finals coming up, a lot of the larger history falls outside my area of knowledge, and a lot of other people can probably attest to it better than I can.

And I guess also to make a short closing point, the fact that a lot of contemporary Buddhism is modernist doesn't mean that it's less valid or "pure" than traditional Buddhism. Many people in Sri Lanka and Myanmar and many Buddhist countries currently follow "modernist" schools of Buddhism, and instead of understanding it as a religious movement imposed mostly by external forces, people like Richard Gombrich will talk about how there is an "elective affinity" between for example the textual fundamentalism of Theravada Buddhism and the sola scriptura of the Protestantism that influenced many Buddhist reformers, or how "modernist Buddhism" really does re-illuminate certain elements of early Buddhism that had become less emphasized over time (while also of course de-emphasizing certain cosmological and social elements of Buddhism).

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Edit: fixing typos and phrasing

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Sources:

The Making of Buddhist Modernism - David McMahan

Curators of the Buddha: The study of Buddhism under Colonialism - Donald Lopez

Theravada Buddhism: A Social History from Ancient Benares to Modern Colombo - Richard Gombrich.