You are absolutely wrong. The whole reincarnation thing probably came from Brahmanism, but Buddha tought impermanence which is absolutely the opposite of any form of reincarnation. There is no soul to be reincarnated.
Yes, there are some Koans and such which use stories involving reincarnation, however these are not literal teachings, they are just stories used to make a different point (usually, something about non duality).
Do some buddhists believe in reincarnation? yep. Some are also christians, or Scientologists, or atheists too. We don't know what comes after we die so it's really up to whomever to think whatever they wish.
We do not teach it, however. And the Buddha did not either. Zen teachers would all agree that reincarnation is a wrong view. Back to Dogen, even. It's literally the opposite of what we think about the nature of self.