Is it right to use the term rebirth in the Buddhist context?
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.