In “New Religions as Global Cultures” (1997, 101-102), Irving Hexham and Karla Poewe write:
“Within the Yogic tradition, ideas of rebirth are logically related to belief in karma. Traditionally, rebirth is explained in two ways. Hindus believe that when human beings die, their souls pass – transmigrate – into another body. Buddhists, in contrast, deny the existence of the soul. They believe that the continuation of sense impressions at the point of death deludes those that suffer near-death experiences into thinking they have an essence or soul. Buddhists postulate reincarnation rather than transmigration. Those who popularize new religions in the West, however, typically fail to convey these distinctions. They tend to define reincarnation as rebirth in human form and transmigration as rebirth in non-human form.”
Feeling muddled? Actually, I think Hexam and Poewe tripped over the language a bit themselves there, even though what they said about the differences in teachings about the soul sounds accurate enough. But my own understanding of traditional Hindu and Buddhist belief is closer to this explanation from Dharmawood:
There is a lot of confusion surrounding the concepts of Reincarnation and Rebirth. At first glance it seems like they are synonyms, but there is a big difference between these two concepts.
The doctrine of reincarnation, also referred to as transmigration, is taught by Hinduism, Jainism and Gnostic Christianity. It’s main postulate is that there is an indestructible, eternal, personal element that travels from one life to the next. This element is called atman in the Hindu religion, jiva in the Jain faith and soul by Gnostic Christianity. Reincarnation assumes identity between the occupant of this body and, when this body dies, the occupant of the following body.
In contrast, Buddhism teaches the doctrine of rebirth. Between a series of lifetimes there is a relationship of causality, not of identity. To make this more clear, let’s use the example of the falling dominos. If I place a series of dominos standing up in line next to each other, and I strike the first one to make it fall down, this causes the second one to fall down, and the next, until the last in line falls down. The fall of the first domino is the cause of the fall of the last one, but there is not a shared identity between the first and the last domino.
In other words, the proper juxtaposition is between reincarnation and rebirth, with transmigration being synonymous with reincarnation; not transmigration and reincarnation, with rebirth being synonymous with both.
Those semantics aside however, I think the really important point is the one Hexam and Poewe make about the distinctions between east and west. That is, that popularizers in the west often redefine the words in ways quite foreign to eastern Buddhism and Hinduism. What is more, I find many westerners also introduce concepts like soul transmigration to or from other planets, angelic intermediate states, progressive evolution of the soul, and other pseudo-scientific and pseudo-Christian ideas that are 100% local.
The upshot for me is, when people talk to me about rebirth or reincarnation, I can never take the language at face value. I assume deeper digging is required until proven otherwise.