Reincarnation does not involve the ordinary notion of "individuals'" personal identity in Hinduism properly understood, only a generalized awareness called 'Atman', which is indistinguishable from the universal awareness called Brahman. The belief in personal identity is decried in Hinduism as one of many attachments that lead directly to suffering and the cycle of Samsara. Buddhism has a very similar understanding, except that they go as far as to pragmatically even discourage any belief in Atman because it's believed that this will not be useful, but will only cause attachment and suffering. "Universal" awareness and liberation from suffering is instead sought mainly as a direct experience via stream entry, which is only one of many avenues in Hinduism and not privileged over others.

There are plenty of folk superstitions in both Hinduism and Buddhism, so it's not impossible that some folks may have believed what you wrote above at some point. But the actual agreed-upon views are pretty clear.