Hi Venerable, 
thanks for replying.
Pardon my ignorace, but I had to google what the AHD is. If I’m correct it’s the American Heritage Dictionary. But that’s an English dictionary, not a Pali one, so it can’t tell us what Pali terms mean. The definitions you’ve given don’t agree with Pali dictionaries like the Critical Pali Dictionary (CPD), Cone’s Dictionary of Pali (DOP), and Rhys-Davids’ Pali-English Dictionary (PED).
Take the CPD for example, which says for kāma: “(mostly in sg.) wish, desire, […] (in pl.) the objects of sensual pleasure viz. rūpa, sadda, gandha, rasa, phoṭṭhabba.” That is to say, the objects of the five senses: sights, sounds, etc. The most natural reading in kāmasaññā would be this plural of “objects” (because otherwise, if we interpret it as “desire”, why would the hindrance of sense desire be emphasized over the other four?) And in the jhana entry formula vivicceva kāmehi (“withdrawn from the kāmas”) we definitely have a plural, which I think the CPD correctly notes are the objects, not “the desires”.
As to limiting the meaning of saññā to recognition or memory, I think that does not agree, well, firstly with the PED which glosses it as ‘consciousness’ and ‘awareness’, but also not with suttas that say you can’t really distinguish consciousness (viññāṇa) from perception (saññā), because the two always go together (along with feeling, vedānā). (E.g. MN43) So going by this, if kāmasaññā ceases, “kāma-viññāṇa” (made-up word, but you get the idea) will cease as well. So I don’t think sañña means a memory-based recognition but just perception in the sense of awareness; at least in this case.
AN9.31, which says kāmasaññā ceases in the first jhana, also say that rupasaññā ceases in the first arupa, (and in the following arupas certain “saññās” of the respective previous arupa cease). I think we interpret the nature of these perceptions very differently, but putting that aside, I don’t see how this can be interpreted as a memory-based interpretation. Do we then forget about the rupa, even though it is still there? To me it simply means that rupa has gone, so you’re not aware of it anymore, you’re not perceiving it anymore. The perception of it has ceased.
Or take the following: “Furthermore, take a mendicant who, with the fading away of rapture, enters and remains in the third absorption. While a mendicant is in such a meditation, should perceptions (sañña) and attentions accompanied by rapture beset them, that’s an affliction for them.” (Sujato tl. AN9.34) I think the most natural reading is that when rapture fades away, you’re no longer aware of it. In other words, you don’t perceive it anymore. If perception (i.e. awareness) of rapture comes back, that’s an affliction because it essentially means you’re back in the second jhana. I don’t think this is about interpretation or memory. By analogy, kāmasaññā also means awareness of some specific thing, which I take to be the objects of the five senses, following the CPD’s interpretation quoted above.
So how do you understand the phrases “withdrawn/secluded from kāma” and “cessation of kāmasaññā”? Excuse me for not having read the entire draft. 
I searched for it but couldn’t find it.
(PS. DN9 also has the cessation of kāmasaññā in the first jhāna. So far that’s 4 Pali suttas. Still unaware if any of these have parallels in other languages.)
Hey Giovanni. Good question. 
First of all this reflection must happen outside of the attainment, because, as you point out, all agree there is no thinking there. So the thoughts that are mentioned can’t happen while you’re in the attainment. (Just like the passage I linked earlier.) But what the DN9 passage exactly means, admittedly is somewhat obscure to me, at least at the moment. It’s definitely a very non-standard way to describe the process, as you can attest by comparing to other suttas. Perhaps @brahmali has an idea?
But for now I think this might be the idea: Interestingly, the terms ceteti (“intend”) and abhisaṅkharoti (“choose” or “generate”) are elsewhere repeatedly connected to the rebirth process. (ceteti eg in SN12.38-40, abhisankaroti eg in SN12.51) Especially the latter, I don’t know a place where it ever describes meditation. So perhaps the passage talks not about what happens in a single meditation session, but about the general tendencies of the mind that need to be overcome. If you still intend towards existence in a form of awareness/perception, you can’t overcome it, you’re still attached. To move beyond, you need to let go of those intentions, i.e. you must become enlightened. And that seems to make some contextual sense because just a few lines later it talks about attaining “knowledge”, which I think we can be quite sure refers to enlightenment. Also, there is this “peak of perception as both one and many”, which may refer to the fact that enlightenment is also a peak of perception, not just deep meditation.
So in short, I think it’s not about meditation per se but about becoming enlightened. At least that’s what I think for now.
Hello again also.
But in my interpretation whatever is described after the jhana can happen after it. For example, as I said above, the second jhana happens after the first, not during it. Yet we continually find the second jhana directly following the first. It means the first jhana has “ceased” and the second jhana is now “in effect”. We also have the passage shared above where there is thinking immediately follows the fourth jhana; but everybody agrees that thinking isn’t present inside the fourth jhana, so it must happen after.
Let me explain in a bit more linguistic detail, then. As @Sylvester explained in a post I linked to before, the phrase “upasampajja viharati” can denote that the jhana doesn’t continue into the following phrases. Although Ven Sujato translates it as “enters and remains in” more literally it says “dwells, having attained”. In other words, you could read it as “dwelling (in general) having attained the jhana”. Or Sylvester reads it as a periphrasis, which is to say the two verbs describe one idea, just like English “I am going to do” has three verbs but the idea is one. I tend to agree with that. The verb viharati often functions more like an auxiliary (“extra”) verb. So the idea of upasampajja viharati is that you have (recently) attained jhana, not necessarily that you are currently still in it (although you perhaps could).
Other readings are also grammatical possible, I won’t discount that, but as I see it you can’t do recollection of past lives inside jhana, so therefore those readings don’t make much pragmatic sense to me.
Hope that helps.