Hi Trevor,
Yes, these words are often used quite indiscriminately. None of these words in their own right implies a “soul,” as far as I can see, but of course it is usage that matters. You may be right that “reincarnation” is often used when a permanent self is understood to undergo the process, whereas “rebirth” is perhaps more neutral. But in the end it is the context that gives these words their distinct meaning.
As for punabbhava it really means “re-existence,” which is pretty much synonymous with rebirth.