Which type of Buddhism believed in reincarnation?

Not all Buddhists believe in reincarnation. The Buddha did not teach reincarnation, although it is part of the doctrine of some Buddhist traditions.

The Buddha spoke of 'rebirth' in some teachings. This is understood by some to refer to a new start in the present lifetime, not the beginning of a new life after death.

The idea of reincarnation may have been absorbed into Buddhism from Hinduism, which does recognize and believe in reincarnation.