Quite close. Cetayamāna is the present participle for ceteti (more on this verb later), which can be rendered as “thinks” or “intends”. Ceteti will help us understand the gradations of intention, using another schema.
Why not all 4?
Leaving aside the vacīsaṅkhāra in SN 47.10 and in the 1st Jhana pericopes, I think the MN 44 scheme can actually accommodate all degrees as suggested. Take a reciter - he/she probably only needs to stop at conceptualisation. A teacher, on the other hand, may need to include formulation (and depending on how sloppy or diligent the teacher is, formulation may need to include constructing the ideas logically, not just syntactically).
And this is where ceteti comes in handy in understanding the different degrees of intention before articulation. Take a look at SN 12.39’s treatment of saṅkhārā in the link to consciousness in the standard DO scheme -
Yañca, bhikkhave, ceteti yañca pakappeti yañca anuseti, ārammaṇametaṃ hoti viññāṇassa ṭhitiyā.
No ce, bhikkhave, ceteti no ce pakappeti, atha ce anuseti, ārammaṇametaṃ hoti viññāṇassa ṭhitiyā.
Yato ca kho, bhikkhave, no ceva ceteti no ca pakappeti no ca anuseti, ārammaṇametaṃ na hoti viññāṇassa ṭhitiyā.
Bhikkhus, what one intends, and what one plans, and whatever one has a tendency towards: this becomes a basis for the maintenance of consciousness.
If, bhikkhus, one does not intend, and one does not plan, but one still has a tendency towards something, this becomes a basis for the maintenance of consciousness.
But, bhikkhus, when one does not intend, and one does not plan, and one does not have a tendency towards anything, no basis exists for the maintenance of consciousness.
(per Bhikkhu Bodhi)
Ceteti is supplemented by its synonym pakappeti, suggesting to me that these are relatively “intentional” or “deliberate” acts of intention/mental kamma of which one is conscious. By contrast, the subtlest form of saṅkhāra is attributed here to the anusayas anuseti-ing. The sense that I get from the suttas is that the anusayas are not even within our conscious experience, unless one were a meditator who’s at least reached the MN 148 stage of sense restraint.
Given that (i) this sutta places ceteti squarely within volitional formations (saṅkhārā), and (ii) saṅkhārā covers the unconscious “intentions” as well, this was the reason I suggested treating the vacīsaṅkhāra as being mere nati (inclination of the mind) in the context of jhana. Saṅkhāra , being the umbrella phenomenon that drives reactions and rebirth (SN 22.79), captures the whole range of “intentions”, from the unconscious ones associated with the anusayas and inclinations of the mind, up to the conscious ones that appear to begin with ceteti. DN 9 addresses the pitfall of cetayamāna in any of the attainments, but this leaves just enough room for unconscious intentions or saṅkhāras to be available in the attainments not to disturb it.
If you think about it, saṅkhārā do persist in the jhanas (MN 64), and they must in fact persist. Without them, there would be, eg. no pleasure born of seclusion. Equally important, what about the reaction of rapture to the pleasure? All these things are the saṅkhata stuff abhisaṅkharonti-ed (fashioned) by the saṅkhārā.
But normal experience should tell you that while you are conscious of your emotions (joy or grief), you can never be conscious of the saṅkhārā that generate them. Grief is generated by the latent tendency to aversion anuseti-ing, but one can be never conscious of such an anusaya, except through inference from the presence of grief.
This is why I feel that DN 9, if read literally by reference to its injunction of ceteti, can allow for much subtler intentions/volitional formations such as the vacīsaṅkhāra to persist. In fact, if you continue reading the passage you cited, what was translated as “choose” is abhisaṅkharoti , also found in SN 22.79, which explains it as follows -
Saṅkhatamabhisaṅkharontīti kho, bhikkhave, tasmā ‘saṅkhārā’ti vuccati.
They construct the conditioned,’ bhikkhus, therefore they are called volitional formations.
It is only by allowing the involuntary and unconscious saṅkhārā to cease without residue that feelings and emotions cease to be generated, giving rise to Cessation. For without feelings, can consciousness stand?