Fair enough, but why the caveat that it needs to be practiced in the east?
First lets look at some Buddhist texts. The Kalama Sutta is perhaps the most famous, but from a lesser known section the Buddha says:
i.e. it doesn't matter if there is a life beyond this one (rebirth), Buddhism is a method developed to help you out either way - the Buddha is explaining an agnostic position on the afterlife.
Here you can also find an argument that some of the earliest Buddhist texts do not teach the doctrine of rebirth:
I hope by appealing straight to the scriptures i have by-passed your need for such non-rebirth believing Buddhists to be from the east. But just in case my anecdote of having met a Buddhist priest who does not believe in literal rebirth (she is Korean, from a Zen school), apparently Shin Buddhism - ironically a branch of pure land Buddhism, and thoroughly eastern - pays no heed to any afterlife.
I accept that the majority of Buddhists believe in rebirth, but the key point is that such belief is not necessary and can (if you want) be taken separate from karma.
There is no reincarnation in Buddhism: any reference to reincarnation is a slip of the tongue. Buddhism generally teaches rebirth - the subtle difference being that there is no transmigration of a 'self'. I think this is explained well here, but there's plenty of online resources.
It is not just the case that Buddhism has a more rarefied and ephemeral concept of self: the concept of anatta, or no-self, is a core Buddhism belief.
That is my least favourite description of Nirvana i have ever seen (for one it's not a place).
My favourite: after enlightenment, the laundry.