Is it right to use the term rebirth in the Buddhist context? As I delve into the Buddhist literature deeper and deeper, particularly the Theravāda sutta literature, I am getting convinced more and more that when we use the term ‘rebirth’, and more so, the grosser English equivalents like ‘metempsychosis’ and ‘re-incarnation’, to signify what the Buddha talked about when he referred to the apparent continuity of the ‘life process’ after death, we are getting it all wrong. I have till now also been unable to find the equivalent of the Vedic term ‘Punarjanma’, which is used so very profusely in the Brāhmanic literature like the Upaniṣads and the Purāṇas. The Sanskrit/Pāli word that comes up again and again in the Buddhist context is ‘bhava’ which has been translated as ‘becoming’, perhaps rightly so. But, would it be right to translate this very word also as rebirth, re-incarnation, punarjanma, and the like, when it seems so very clear that bhava does not stand at all for any of these, because the very notion of rebirth/re-incarnation/punarjanma carries within it the concept of a permanent entity moving from birth to birth?
I remember a beautiful metaphor from somewhere that compares the notion of this recurrence of saṁsāra in Hinduism with that in Buddhism. If this recurrence is like a necklace of pearls in Hinduism where the pearls stand for various janmas and the string for the eternal ātman, in Buddhism it is like a pile of coins where each coin, each birth, thought dependent for its support on the coin below, on the birth that came before, does not have any eternal binding entity holding them together, only the unseen ‘gravity’ of karma. Isn’t it right, therefore, that the actual term in the Buddhist sense, used for this recurrence of lives, ought to be bhava, or, to be technically more exact, punarbhava/punabbhava/’re-becoming’/recurrent becoming, rather that punarjanma/rebirth/re-incarnation/metempsychosis?
It is very clear that the english word 'rebirth' in Buddhist writing and practice causes a whole host of confusions and misunderstandings. This is evident in this forum with the myriad questions and debates that have erupted as a consequence. As an added complexity, not all of the confusions and misunderstandings are related or easily dispelled in the same way.
Now, with that said I don't quite know how to answer the question, &quot;Is it right to use the term rebirth in the Buddhist context?&quot;
Let's grant the supposition that the answer is yes and see what the consequences are:

The word 'rebirth' could be replaced with another english word or words
The word could be omitted entirely with a blank in its place

The problem with #1 is that this has already been tried. In fact, numerous other words have been suggested instead. You've mentioned two - metempsychosis and re-incarnation - but many more have been used in the past with transmigration being another heavily used synonym.
Some on this forum are heavily biased against the word 'rebirth' thinking it causes more confusion than not and have insisted that 'rebirth' was never uttered by the Buddha. Which is true insofar as the Buddha did not speak english.
For me, I see all of this arguing over the 'right' word to use as rather beside the point. I don't think it is true that there exists one or more 'right' words that would magically dispel all the confusion and misunderstanding that come along with the word 'rebirth' or the ideas that the word is used to convey.
My go to strategy to try and dispel the confusions and misunderstandings that the word 'rebirth' and the ideas behind it often inspire is to focus not on 'rebirth' from life-to-life, but rather 'rebirth' from moment-to-moment. Why? Because I've found that most Buddhists or people learning about Buddhism have an easier time thinking about and granting that 'rebirth' from moment-to-moment happens and can relate it back to their own daily moment-to-moment lives in an experiential way. And a lot of the same questions that pop-up in 'rebirth' from life-to-life also pop-up in a natural way in 'rebirth' from moment-to-moment.
Let's take your contention that, &quot;the very notion of rebirth/re-incarnation/punarjanma carries within it the concept of a permanent entity moving from birth to birth?&quot;... that is true of 'rebirth' from moment-to-moment as well isn't it? Don't we all have the sense and experience that there is some permanent entity moving from moment-to-moment? So if we can resolve how it can be that 'rebirth' from moment-to-moment happens all the while not having some permanent entity moving from moment-to-moment, - and it most undoubtedly does happen as we can all attest experientially, right? - then perhaps we can take that and extrapolate what it means for 'rebirth' to happen from life-to-life.