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

All possible birth is truth either a smallest moment or reincarnation, but the reality truth is the smallest particle which can see after getting strong enough concentration meditation and insight meditation. This is the real birth which imagined by the ordinary as imagination truth or imagination fake. It something like we think atom is car but it is atom actually, but the reality is the smaller, the faster, unique, and complicated causing each others than atom which we have known.

Birthing reality is depending various causes, so birthing reality is uncontrollable, none self. When the practitioners' concentration meditation and insight meditation are strong enough, they are going to see the truth that...

1. SantatiGhana: Birthing reality is arising and vanishing more than trillion moments a second because of causes, but ordinary person imagine they all are same long living for 1 minutes or 1 days or one life.
2. SamūhaGhana: Birthing reality is so small than atom or particle (paramāṇū's qualities), but ordinary person imagine they all are same one loaf for 1 minutes or 1 days or one life or forever.
3. KiccaGhana: Each birthing reality has difference jobs/specifications/qualifications when they causing their effect together, but ordinary person imagine they all are same doing/doer for 1 minutes or 1 days or one life or forever.
4. ĀrammaṇaGhana: Each mind and mind factors process has difference object, but ordinary person imagine they all has same object for 1 minutes or 1 days or one life or forever.

So, birthing reality is uncontrollable, none self. And then the practitioner meditate above view with the other wholesome practices, then 3 characteristics are going to appear clearer and clearer. The whole world appear as vanished every smallest moment. It's good to see like that, no fear because it is truth either we can see it or not. This seeing is leading to finish the whole broken world.

One of the difficulties of translation is that it operates on multiple levels. The translation of an individual word often depends on the meaning of the sentence it is embedded in, the meaning of a sentence depends on the nature of the text it is embedded in, and the nature of a given text relies on subtleties of the author's worldview. This is particularly difficult with spiritual and philosophical texts, which are written from esoteric worldviews, in ancient or specialize languages.

The point is that we ought to carry words lightly until we can look back and understand them correctly; this calls for a deep level of translation, not a shallow investigation of the meaning of words.

But in the spirit of the question, the best English word to use here would probably be the (somewhat arcane and obscure) term 'recrudescence'. Recrudescence means "the recurrence of an undesirable condition" (literally "the return of the raw"): it's most commonly used with a disease, in that a disease which is not completely eradicated will hide out in odd corners of the body or of the world, and burst out again when conditions are appropriate for it. We might say, for instance, that shingles is a recrudescence of chicken pox, since the varicella virus that causes chicken pox in children lies dormant in nerve tissue until the decades-long immunity fades and it can replicate again in older adults.

In the Buddhist sense, the 'unpleasant conditions' are states of tanhā (craving and attachment), which constantly resurface and recur until the root cause is removed. The notion of reincarnation arises because (again, on a somewhat shallow philosophical level) many people presume that the egoic self is the root cause of tanhā, and so the egoic self must itself be recurring, and will continue to recur across lifetimes. But as I see it, the egoic self is epiphenomenal, not causal — the sense of 'I' as an object is a result of our attachments, not the source of it — so there is no intrinsic egoic self to be transferred. Instead, we have an epidemiological situation: our 'unpleasant conditions' don't just recur within ourselves, but spread and take root in others. The tanhā that forges a sense of 'I' (an 'I' which we hold so intimately and personally) can spread through and lie dormant in society, awaiting a moment to break out as 'I' again. We don't reincarnate in any personal self; we create the conditions in which egos of the sort we hold close to ourselves can recrudesce in other times and places.

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

Let's grant the supposition that the answer is yes and see what the consequences are:

1. The word 'rebirth' could be replaced with another english word or words
2. 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, "the very notion of rebirth/re-incarnation/punarjanma carries within it the concept of a permanent entity moving from birth to birth?"... 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.

I will comment more later and provide sutta quotes. Initial points:

1. As said in the question, there appears no commonly used equivalent to punarjanma (puna-jati) in the Pali, apart from "dukkhā jāti punappunaṃ" found in Dhammapada 153. However, the meaning of "dukkhā jāti punappunaṃ" depends on the meaning of the word "jati". It is very clear by the Pali suttas as well as commentary such as the Visuddhimagga the word "jati" has numerous different contextual meanings and ultimately does not have the core meaning of "reincarnation", let alone "physical birth". The simplest example is the use of the word "jātiyā" in MN 86; kiṃjātiko found in many suttas; or the common use of the verb "jāyati" to refer to coming to be of mental phenomena, such as love, sorrow, rapture, etc.

2. In the words commonly translations of "rebirth", the words "jati", "janati" or "jayati", which have the root "jan", are generally not found. Instead, the words commonly translated as "rebirth" are based in the root "pad" rather than "jan".

3. However, there is the word "paccājāyati" (which does not include "puna") however this is rarely used and an investigation of it may find it does not mean "reincarnation" but, instead, "reclassification". "Paccājāyati" appears to be found in the context about the status of a "jati", such as "human", "god", "ghost", "animal", etc, rather than is used to describe the destinations or results of kamma, the later being by far the most common context where the translation "rebirth" is used. Here, in respect to kammic inheritance, words rooted in "pad" (such as "upapajjati", "upapanna", etc) are most commonly found. Note: MN 148 clearly shows the word "upapajjati" does not literally have any connection to "reincarnation" but merely means "to follow from the former".

4. As for the word "bhava", this is one of the three "asava" ("mental outflows"; together with sensuality and ignorance). It is obviously very wrong to say "bhava" means "reincarnation" because "bhava" is simply and literally a state of mind. For example, MN 44 says "bhava" is a cause of "identity". MN 121 says "bhava" is a "perception".

Good question, well thought out and described in detail. Your proposal 're-becoming' may be better than the other words, at not implying an immutable soul underlying, compared to 're-birth' and 're-incarnation', 'transmigration', etc.

But it has its problems as well. The biggest one being that it doesn't easily convey you're talking about rebecoming after a physical death.

In the end, it comes down to what people agree on for a convention, or official dictionary definition. Theravadans usually prefer 'rebirth' over 'reincarnation' and the other words. Originally in the dictionary, 'rebirth' didn't even have to do with physical death and what happens after. That's probably why Theravadins chose 'rebirth' over the more widely used 'reincarnation', and other words.

Now that the Buddhist idea of 'rebirth' is in some dictionaries, it seems to be more commonly understood in Theravadin Buddhist context does not entail a permanent soul underlying. Other religions though, like Hinduism, use 'rebirth', 'reincarnation', with a soul/atta.

As long as the convention ('rebirth' in this case) is basically well understood, it's best to stick with it, otherwise you'll waste a lot of time having to explain to people what your new word means exactly.

The time to abandon words, is when it suddenly achieves wide adoption with a completely different and wrong meaning (compared to the original). For example, 'gay' used to mean 'happy and joyful', but once the common definition became 'h*m*sexual', then you have to respect convention and change to a different word or risk being misunderstood.

Dhammapada 153 - 154 in Pali:

Anekajātisaṃsāraṃ,
sandhāvissaṃ anibbisaṃ;
Gahakāraṃ gavesanto,
dukkhā jāti punappunaṃ.
Gahakāraka diṭṭhosi,

puna gehaṃ na kāhasi;
Sabbā te phāsukā bhaggā,
gahakūṭaṃ visaṅkhataṃ;
Visaṅkhāragataṃ cittaṃ,
taṇhānaṃ khayamajjhagā.

Translation of Dhammapada 153 - 154 by Ven. Buddharakkhita:

153. Through many a birth in samsara have I wandered in vain, seeking the builder of this house (of life). Repeated birth is indeed suffering!

154. O house-builder, you are seen! You will not build this house again. For your rafters are broken and your ridgepole shattered. My mind has reached the Unconditioned; I have attained the destruction of craving.

Repeated birth here is "jāti punappunaṃ".

From Ven. Nyanatiloka's Pali dictionary entry on "jāti":

jāti：'birth'，comprises the entire embryonic process beginning with conception and ending with parturition．

＂The birth of beings belonging to this or that order of beings，their being born，their conception （okkanti） and springing into existence，the manifestation of the groups （corporeality，feeling，perception，mental formations，consciousness； s．khandha），the acquiring of their sensitive organs：this is called birth＂ （D．22）．For its conditioning by the prenatal kamma-process （kamma-bhava； s．bhava），s．paṭiccasamuppāda （9，10），paṭisandhi．

One of Wisdomlib's entries on jāti states:

Jati or jata means arising or coming up.

The Dhp 154 commentary by Ven. Buddharakkhita states:

According to the (traditional) commentary, these verses are the Buddha's "Song of Victory," his first utterance after his Enlightenment. The house is individualized existence in samsara, the house-builder craving, the rafters the passions and the ridge-pole ignorance.

From these, and from other references on clinging aggregates (SN 22.48) and Nibbana-element with residue remaining (Iti 44), my take on this is that "birth" simply means the arising of one's individuality (the self, the music of the Vina Sutta), based on the operation of the five clinging aggregates (the lute) working together in the way described by dependent origination.

And repeated birth refers to repeated occurrences of this, sustained by craving as described in SN 44.9.

Ven. Buddharakkhita's commentary above supports this.

Another interesting fact is that in some suttas like AN 5.28, "past lives" is the usual translation for "pubbenivāsaṃ", which literally means "previous homes". "Homes" and "houses" (from Dhp 154) sound similar, don't you think?

In suttas Buddha uses two kinds of language. One language employs worldly concepts such as rebirth. Another language introduces technical concepts such as Dependent Origination.

The first type of language is very simplistic and is meant for beginners, the technical language is much more precise and is meant for advanced students.