Hey Ajahn,
At the very least you are saying there is also some sort of bhava devoid of kammic processes, then? But if bhava in Dependent Arising primarily meant some karmic activity, then the cessation of that type of bhava would happen in this life, and arahants would no longer have it. It’s a bit artificial to divide bhava into two types, one that applies to an arahant, and one that does not. Nothing in the suttas really suggests this. They just say bhava ceases at parinibbāna. (Iti44)
And how would the no-kamma bhava of the arahants fall in the explanation given in AN3.76-77, if bhava there includes kamma, as you suggest? Arahants don’t make kamma, so their bhava isn’t included in this sutta, then? In my opinion, their bhava is included, because they live in one of the three realms mentioned. It’s simply the bhava that started when they took rebirth as a result of kamma (and craving) in the past life.
When bhava is actually (kind of) defined, the definition is indeed simpler, and exactly that of life in different realms. But that happens not here, but in suttas such as SN12.2: “And what, bhikkhus, is existence? There are these three kinds of existence: sense-sphere existence, form-sphere existence, formless-sphere existence. This is called existence.”
I don’t think AN3.76-77 are definitions. For starters, they would be much more clumsy and longwinded than the usual definitions we see in the suttas. 
I suppose it hinges on the word kittāvatā. Ven. @Sujato translates it as, “How is continued existence defined”. But “defined” often this is not the idea of the word, if ever. (Edit: It depends on how we interpret ‘defined’. Venerable Sujato responded here, and made me change my mind somewhat about kittāvata. I still don’t think it is a definition as I initially interpreted it. But my suggestion below of “how” was going too far the other way. I left my post unchanged, though, because my general thoughts remain the same.)
Take SN12.51 for example: "When a bhikkhu is making a thorough investigation, in what way (kittāvatā) should he thoroughly investigate for the utterly complete destruction of suffering?” Or let’s say “how should he thoroughly investigate?” Now let’s consider AN3.76 with this in mind. I adopted Venerable Bodhi’s translation, but with ‘how’ for kittāvata instead of his ‘in what way’.
“Bhante, it is said: ‘existence, existence.’ How (kittāvatā), Bhante, is there (hoti) existence (bhava)?”
In other words, how does existence happen? How does it come about?
The verb hoti is already awkward for a asking for a definition of a word, which is otherwise done with nominal phrases (without a verb) and using katama, such as Katamo ca, bhikkhave, bhavo? (SN12.2) Or one with the ‘vuccati’: ‘Arahattaṃ, arahattan’ti, āvuso sāriputta, vuccati. Katamaṃ nu kho, āvuso, arahattan”ti? (SN38.2) (Also, questions with the ‘bla, bla, āvūsu, vuccati’ are far from always definitions. See e.g. SN22.21: “Sir, they speak of ‘cessation’. The cessation of what things does this refer to?”.) So, to me hoti asks about how bhava comes to be (or ‘becomes’), which is not an uncommon use ofhoti. And the answer that follows is in line with this, not in line with a definition. So continuing the sutta:
“If, Ananda, there were no kamma ripening in the sensory realm, would sense-sphere existence be discerned?”
We can also translate the verb paññāyetha as “would … occur”, like Ven. Sujato. That’s the implication of the verb in other contexts too. It connects better with the question, “how does existence happen?” The idea is that existence can’t occur without kamma of a past life ripening in the next life. Without kamma that leads to rebirth in the sensory realm, no-one would be born there, and there would be no existence (bhava) there. The text continues:
“No, Bhante.”
“Thus, Ananda, for beings hindered by ignorance and fettered by craving, kamma is the field, consciousness the seed, and craving the moisture for their consciousness to be established in an inferior realm. In this way there is the production of renewed existence in the future.
The Buddha’s answer to Ananda’s question, “how is there existence?”, ends with, “in this way there is the production of renewed existence (punabbhava) in the future.” This is not a definition, but indeed an answer to how existence comes about—an answer very much in line with all the other discourses. (Unlike the idea of bhava = kamma, which we don’t find anywhere else.) The Buddha answers that bhava (or “renewed existence”, punabbhava) is produced by craving and kamma of the past life. Consciousness being established in a realm refers to rebirth too; it’s consciousness moving on from one life to the next.
This idea of bhava meaning just existence, not kamma, aligns with all the references I started this topic with. For example, MN127 mentions “rebirth in a bhava” (bhavūpapattiyo), which it then explains happens after death, not while alive. Also take AN4.131, bhavapaṭilābhiyāni saṁyojanāni ‘fetters for getting an(other) existence’, which to me makes little sense as “the fetter for getting kamma” or however that would be interpreted in light of the commentary’s idea of kammabhava. This is the sutta on the “in between” lives, which also indicates the bhava one is “fettered to” is that which happens after death, not something that happens while alive. And so forth; see the references I have given when you have the time.
And talking about bhavataṇhā. I would make almost the exact same point you made before, which was good: We crave existence, not some sort of kammic processes within existence, or whatever. And rebirth is the core idea here too, since bhavataṇhā is a form of taṇhā ponobbhavika, ‘craving that leads to rebirth’. Bhavataṇhā is craving for continued existence after death, not craving for something in this life.
I really think the idea of bhava including kamma fits none of the uses of the word bhava in the suttas. It’s the craving and upādāna that are the kammic processes, not the bhava you crave for or take up (upādāna). The idea of bhava in the suttas is much more simple and natural than that of the commentaries. It just means existence as we would use the term in English, not some sort of karmic activity. Bhava is something passive in the suttas, not something active. To include in bhava activities done in this life (kamma) actually has many of the same problems I see with “becoming”, since it’s a movement towards a one-lifetime interpretations of Dependent Arising. (In fact, I wouldn’t be surprised if it was exactly these kinds of commentarial ideas that initially gave rise to the translation “becoming”. Like in Ñaṇamoli’s tl. of the Visuddhimagga, where it is translated ‘becoming’. But in the commentaries that translation works a tad bit better than in the suttas, because it is exactly this context of kammabhava where ‘existence’ doesn’t really work as well, if you ask me.)
I know this interpretation removes some of the daily-life applicability of how you’ve explained DA before. But then DA is an explanation of the second noble truth, which is the craving that leads to rebirth (taṇhā ponobbhavika), not any ordinary type of craving. The craving in DA is for all intends and purposes also taṇhā ponobbhavika, and therefore the bhava that follows craving refers to a next life too, just like in the noble truth. The links of DA from craving till existence present the same idea as the second noble truth, but it uses bhava instead of punabbhava. A minor difference, really, ‘existence’ versus ‘again existence’. The word ‘again’ is effectively implied by context in DA, since bhava there leads to birth.
See also Snp3.12:
Upādānapaccayā bhavo,
Bhūto dukkhaṃ nigacchati;
Jātassa maraṇaṃ hoti,
Eso dukkhassa sambhavo.
Dependent on taking up, there is continued existence.
Having come to be [in the next life], you undergo suffering.
Being born, you will die.
That is how suffering comes to be.
The word bhūta also often refers to birth. Here bhava is best taken as referring to this “coming to be”, i.e. it means continuing to exist after death. It’s again not about kammic processes.
Or consider MN75:
And [when you become a stream enterer] you will think: I have long been tricked, cheated, and deceived by this mind. For what I have been taking up [at rebirth] was just form, feeling, perception, will, and consciousness. And dependent on that taking up of mine, there was continued existence.
The aggregates “start” at birth, which is here effectively equated to bhava. The taking up of the five aggregates results in continued existence after death. Not in some kamma activity.
Your quote has no connection with Dependent Arising or bhava. It’s another context altogether, about starting fires. But the suttas I mentioned are about DA. And here these words are about rebirth. I mean, abhinibbati (the noun) is part of the definition of birth, and in DN15 it is used for the embryo arising in the womb. It could hardly be any clearer, I would say.
I’m very confused by all this. Of course I don’t know what everybody throughout all of time has written, which is why I specifically said “generally”. I was also under the impression that Ven. Buddhadasa wrote in Thai, not English. And what does he have to do with the Visuddhimagga? If anyone departs from the Visuddhimagga it was him, not me. Anyway, I’m not concerned with what Buddhadasa said, but by what the Buddha said. 
Iti44 says exactly that, that bhava only ceases at parinibbāna: