e.g./followup: how is Buddhism a religion? (uncreated universe, no God)
I can't really answer the main question. My guess would be that every historian who has to deal with the topic in detail probably operates from a slightly different definition, while other historians deal with more traditional understandings of religion. (I.e. I'm currently looking at The Heart of Europe which deals with the Holy Roman Empire. It pretty much takes standard ideas constituting Catholicism, Protestantism, and Judaism. No real surprises or a lot of nuance regarding them, AFAIK.)
But on the question of "how is Buddhism a religion?" I can help.
First off, Buddhism very much does have Gods. The earliest biographies and stories of the Buddha reference him in discussion and being supported by "the gods" and the Buddha never declared himself an atheist or materialist. In fact, this would be one of the extremes the Buddha's Middle Way specifically guarded against. Buddhism has no creator God, in that it is rather agnostic (strictly speaking, apathetic) to the origin of the universe and how it was created. That said, every traditional Buddhist tradition - from India, to China, to SE Asia, to Tibet, to Korea, to Japan - has a pantheon of deities. Followers of these traditions take a wide spectrum of views regarding what exactly "god" or "gods" means in the context of their worship. A generally common view in Tibetan Buddhism is that the gods are objects of worship for those at the lowest level of spiritual maturity, that they are object lessons and mystical portrayals of enlightenment for those higher up on the chain, and finally that they are illusory manifestations of the self to be either cultivated or cast away (i.e. the gods are inside you) at the highest level of spiritual attainment.
The Sanskrit word for "god" is deva and refers to a class of beings in Buddhism. The devas are wholly different in form and in kind from the Abrahamic God (in its modern form). Devas are not omniscient, omnipresent, or omnibenevolent. Probably strangest to those of us coming from Western traditions, Buddhist gods are not even immortal. Insanely powerful by human standards (in their own realm) and very long lasting, but once the karma from their previous lives runs out, the gods will decay (die) and descend into lower realms of existence. This is the main reason the gods praised the Buddha, and why there are stories of Buddhas ministering to gods: because they are susceptible to the same cycle of birth, decay, death, and rebirth as all living beings. The only way away from suffering is not to become a god, but to transcend existence itself.
Though I want to be clear: the gods of early Buddhism, drawn from the culture of ancient India, and the gods that exist in today's Buddhisms, drawn from the cultures of their surroundings, and the lack of gods in the western wave of Buddhism (or as some might argue "pure Buddhist logic") doesn't equal or not equal Buddhism being a religion. If the necessity of faith in a deity (regardless of that definition) is what a religion requires to be a religion, then we can award Alcoholics Anonymous the status of "Religion" and demote Confucianism and Taoism. It starts to get ridiculous real fast.
Damien Keown in his A Very Short Introduction to Buddhism spends the first section of the book over what he calls "The Seven Demensions of Religion" particularly to point out that Buddhism (and while we're on the subject, Taoism and Confucianism, though that's obviously not the point of the book) has everything that traditionally defines religion without the centrality of an all-powerful deity.
The Seven Demensions are:
I think they largely speak for themselves, but I'd be happy to elaborate specifically on aspects of the Buddhist tradition regarding each if you have pointed questions on any of them in particular. The main point Keown makes in the book is that Buddhism fulfills each criteria where we would expect Christianity, Judaism, and Hinduism fulfills also. Whereas other traditions (say, for example, culinary, or literary, or Christmas) would fail at least one or more.
Those who usually point out that Buddhism has "no god/s" and therefore "can't be a religion" has never experienced life in a Buddhist monastery or temple. Traditional Buddhists have a lot of faith in their deities, regardless of the philosophical and theological relationship to the Buddha. And traditional Buddhist histories place a lot of emphasis on the relationship between their leaders and (broadly speaking) divine authority. Two of the most famous flights out of Tibet, the Zhabdrung's in 1616, and the Dalai Lama's in 1959, were prefaced by consultations from deities. The Zhabdrung had a dream in which the guardian deity of the Kagyu tradition, Mahakala, fled southwards to Bhutan. In the Dalai Lama's case, he consulted the famous Nechung Oracle which warned him to "go now" and even drew a helpful map to India. Korean Buddhist temples are particularly cross-religious, mixing Buddhism with Taoist belief systems and local (usually mountain) deities (hence the name: San-shin). Buddhism served as a cultural vessel, bringing Indian gods to the fringes of the Buddhist realm (the Guardian Kings of the Four Directions is a perfect example) while also adopting local traditions and gods to its belief structure. I've written much about this phenomenon in Tibet/Himalayas here.
Anyway, if Buddhism isn't a religion because strictly speaking it does not require a god at the center of it, then Confucianism and Taoism aren't either. Unless one extends the definition of "god" so far to include an amorphous unknown on one end, and your Dad on the other end.
Sources:
A Very Short Introduction to Buddhism by Damien Keown
Buddhacarita, or Acts of the Buddha by Ashvagosha
A Concise History of Buddhism by Andrew Skilton
The History of Bhutan by Karma Phuntsho
Tibet: A History by Sam Van Schaik
An Encyclopedia of Korean Buddhism by David A. Mason
I like this definition from Clifford Geertz:
A religion is a system of symbols which acts to establish powerful, pervasive, and long-lasting moods in men by formulating conceptions of a general order of existence and clothing those conceptions with such an aura of factuality that the moods and motivations seem uniquely realistic.