Is it true that the caste system is the fault of the British?

No. The caste system was merely the name that the British assigned to the system of social organisation that they observed in India. To say that it was somehow shaped or manufactured by the British would be to ignore the previous 3000 years of history which shaped the caste system which the British came into contact with in the 17th and 18th centuries.

Strictly speaking the manner in which most scriptures define caste, the caste system or its development and evolution, has little to do with how castes and caste hierarchies actually developed in the real world. For example, texts such as the Purusha Shukta may describe how the "cosmic being was divided into 4 parts and out of each part namely the head, the arms, the torso and the legs, each of the varnas were formed namely the Brahmins, the Kshatriyas, the Vaishyas and the Shudras respectively". However this implies two things, 1) That the caste an individual was born into dictated the occupations he was allowed to choose in life and that throughout history these were the only occupations said individuals chose. 2) That there was a religious justification and foundation for caste first and its real life implications second. However this isn't true. The distinction between the upper or elites in Vedic society and those who would serve as servants or slaves was made in the Rig Veda itself. The Indo-Aryans referred to themselves as Arya or Aryans while they referred to the native inhabitants of the subcontinent as "dasas" literally meaning servants. This relationship between ruler and ruled and strict class and race based distinctions were not unique to Indo-Aryan society. Now, while there was almost a millenia during which heavily male dominated groups of Indo-Aryans took native female partners as well, the idea of the purity of the patrilineal bloodline existed. Meaning while it was acceptable for a elite male to take a dasa female partner the opposite could not be acceptable. It was this patrilineal bloodline that is the foundation of what is known as gotra. A person's gotra is their patrilineal bloodline. Only the three "upper castes" are assigned or have gotras. The Shudras do not. In later centuries the acceptable form of marriage in society would be called "Anuloma" marriages and apart from marriages within the same caste which were considered excellent, this would be the only form of marriage deemed at least acceptable.

With the passage of time, the Indo-Aryan or Vedic society divided itself into 3 upper and 1 lower varna. Namely Brahmin, Kshatriyas, Vaishyas and Shudras. The first three being born out of the fact that occupations and their know how was passed from fathers to sons and hence occupations themselves became hereditary. Overtime, religious justifications for this hereditary system were created. There is also the idea that one could fluctuate between varnas, which is also unfounded in history.

So, caste, or. Jati is as old as 2000 years give or take, and the foundation for this system of differentiation between groups of people was the perception of the Indo-Aryans towards the natives of the lands they came to settle upon and the relationship between a ruling elite and it's servant class/ethnic group.

As for sources on the topic I recommend :

"India's Ancient Past" by RS Sharma

"A History of Ancient and Early Medieval India" by Upinder Singh

They're quite comprehensive, especially Upinder Singh. But RS Sharma is a must read.

u/MaharajadhirajaSawai has written already really nicely about the ancient origins of caste in India, and I think it's sufficient to say that the British did not invent caste. But I also think it's worth addressing the question u/Dragon9770 poses about how the British impacted ideas of caste. I've heard variations of this question before, and I think well meaning campaigners against the strictures and inequities of the modern caste system have found it useful to suggest that caste is somehow 'un-Indian' and that India's modern jati are the result of its colonial oppression. A democratist would find this an appealing idea, I think. You can see how that might prove persuasive, as, if true, it puts the challenged party in a dilemma: do they prefer to say that they are happy about endorsing a tenet of imperialism, or do they consent that the jati are an outdated social structure? Of course, it's a false dichotomy, but it's actually quite muddy water I think. What did the British think about caste, and how did this affect their governance?

As u/MaharajadhirajaSawai notes, the British observed this extant social structure, and clearly did not invent it sui generis. To say that the notion of caste was status quo ante in 1947 vis à vis the Company Raj period would not be accurate, I think. The Company Raj period can be characterized in its latter days as having a tendency to want to 'civilize' and export British ideals to the subcontinent, with Christian missionaries hard at work and the states goal of the East India Company to replace corrupt and despotic regimes that were ill serving India. Post-1857 policy in India differed in an important way. The Indian Rebellion[1] brought about the period of Crown rule in India from that year, with British India at that point being an officially colonial possession of the British Crown and governed ultimately through Westminster parliament. This period is defined by a big shift in policy towards India. Missionaries were restricted in their work where it was most disruptive to local life, and old structures were to be preserved and honoured rather than disrupted for the sake of 'Europeanizing the natives'. This policy change can be briefly explained by two motivations: financial and idealistic.

For the first motivation, many historians have identified how the British government was concerned to preserve India's profitability. BR Tomlinson writes[2] that the higher levels of metropolitan Britain's government had actually quite narrowly defined requirements for imperial policy in India, and sought to bring these about through political action. They were keen to keep India profitable for Britain, to keep it capable of paying down its debts and supplying a military contribution to the effort of protecting the Empire. Britain had struck metaphorical gold in India with the textile trade. The profitability of textiles had exploded with the industrial revolution and the invention of factory production. Some historians have argued how India was kept purposefully deindustrialized in order to fuel a trade triangle: India represented an enormous source of primary resource to fuel the textile mills of Great Britain that could never have met their needs domestically. By the trade rules enforced in India, the subcontinent also then represented an enormous market to sell cloth back to.[3] The Mutiny had shown how fragile this wealthy production line could be, and parliamentarians and industrialists alike were keen not to let trade be disrupted by discontent actors in India. The Mutiny was famously sparked by a failure to respect cultural dietary strictures, and represented in microcosm a problem being encountered all over India; where British values or cultural ideas were being inserted into Indian life, there was resistance to greater and lesser extents. This resistance could be minimized by respecting the stated cultural preferences of Indians. For the second motivation, where Company rule was characterized by religious, cultural, and even political missionaries, we can see in the post-Mutiny period British campaigners seeking to preserve India's ancient culture. In 1858 Queen Victoria made a proclamation of respect for Indian customs, and for Indian princes. Edward Said's Orientalism talks about how the West viewed the East as a mystical place of exotic 'otherness', fundamentally different from the way societies were formed in Europe. Where they did not try to crush this culture, instead they elevated its status to one of even reverence. A chair of Sanskrit professorship was created at the University of Oxford, elevating it to the status of Latin and Greek as one of the classical tongues. Just as classical writers were sources of timeless knowledge, so could the ancient Vedas be too. This is the India that Kipling and others wrote about and popularized.

In either case, if the British were to safeguard what they considered essential to India's character, they needed a way to understand it and quantify it. The first British census of India occurred in 1872, and by the end of the century included questions of caste. A census was a scientific way to understand and quantify a place, and would tell the British administrators what they were working with. The British understood that India was made up of castes of Hindus, as well as a large minority of Muslims and Sikhs, and a smattering of others like Zoroastrians.[4] In order to make sure that the needs of a cultural group were being met (whether they met these needs is another question for another time) they needed to define it. So a British official would go toall the towns, all the cities, across the territories, and have all the locals submit to the census. Perhaps most importantly they'd go to the villages, seen by the British as the fundamental essence of Indian culture, where India at its purest ancient representation still resided. Each local would be asked what caste they were. Landowners would be identified. Even where their 'ownership' of the land was formerly a loose and informal notion, now it became concrete, scientific, and official. By this process, a structure could be built within an imperial hierarchy that made sense to the British, with landowners sitting atop labour castes sitting atop untouchable castes. Princes above them all, and the British Sovereign above them.

So the British concretized these relationships that had previously existed in a codification of something that had previously been to a certain extent implicit. But they also enforced categorization where previously lines had been unclear, or had not existed. The British understood India to be 80% 'Hindu'. We know today that to be Hindu is not the same concrete notion as, for example, to be Anglican, or Roman Catholic. India had not ever been a homogeneous religious monolith. Landowning castes had different names in different regions. Different castes did different jobs in different regions.[4] India's languages are a patchwork, and its social castes often flowed into one another, with hierarchies strengthening in places, weakening in others, and appearing in different orders of hierarchy in yet others. Martial jati could be higher or lower on the scale depending if they were from one region or another. In their own way, groups like Sikhs and Muslims were not outside this structure. In some regions, Muslims were the poor, labouring majority.[5] In others they were the wealthy administrators and lawyers. By taking India as a whole and considering that it needed to be quantified in this rigid way, the British enforced a rigidity on it. By taking photographs (another new science), the British identified what the 'correct' traditional garb or appearance of certain groups was, where previously no reference point existed to conform towards.[6] So by seeking understanding, the British also transplanted their own understanding and their own values.

In conclusion I think it would be fair to say that the British did enhance the notion of caste, and make it perhaps more tangible than it might have been to some. And their policy was not without contradiction: just as they worked to preserve this structure, they also acted upon their own moral instincts and took actions regarding the legal age of marriage, suttee, and similar, which was often not popular with the communities the laws affected. Also worth noting is how many Indians atop this social hierarchy had a vested interest in preserving and enhancing it. The British administrator seeking clarity on a point of caste would not be chinwagging with a peasant farmer or muck-cart worker, but would ask his chum the lawyer, or even prince, who was likely to give an answer that did not unseat their lofty position. So it was also a collaborative process. And it is important not to ascribe malicious agency where it did not exist. The "fault" of the British assumes they went into this project with a design, which would be a stretch of the imagination to believe I think.

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[1] The name of this conflict is disputed territory with politically charged overtones, and is also called the Indian Mutiny and the First War of Independence.
[2]BR Tomlinson, 'The Political Economy of the Raj: The Decline of Colonialism', The Journal of Economic History Vol. 42, No. 1, The Tasks of Economic History (Mar., 1982), pp. 133-137
[3] An overview of the relationship can be found in the Cambridge Economic History of India
[4] Nicholas Dirk, Castes of Mind: Colonialism and the Making of Modern India, Princeton University Press
[5] Peter Gottschalk, Religion, Science, and Empire: Classifying Hinduism and Islam in British India, OUP
[6] Christopher Pinney, Camera Indica: The Social Life of Indian Photographs, Reaktion Books