When I want to analyse a sutta or a translation there are two places I turn to.
One is Ven. Sujato's translation with the "View root text with translation" and "Activate Pali word lookup" options enabled.
That gives this:
The Buddha has spoken of not identifying even with the attainment of the first absorption.
‘paṭhamajjhānasamāpattiyāpi kho atammayatā vuttā bhagavatā.
For whatever they think it is, it turns out to be something else.’
Yena yena hi maññanti tato taṁ hoti aññathā’ti.
I think that's more like the first translation you quoted than the second -- i.e. it explicitly references the first jhana.
The second one you quoted gives you the impression that it might be a more general statement, presumably because it says "in whatever way they conceive" and not "whatever way they conceive it".
Ven. Sujato's translation says explicitly "it" (referring back to the previous sentence); and I think that's justified by the word tato.
I'm not denying that atammayata isn't a good thing generally, but to answer your question I think that here the object is specifically the first jhana.
But the English translation says literally, "For whatever they think [the first jhana] is" -- however perhaps a different translation might be, "For whatever opinions they form [about anything] from the first jhana" ("from" being a translation of tato), which tends again towards that second (more general) meaning.
The other place I go is Piya Tan's analysis (if there is one); partly because he references other writers, and commentary, and so on.
His analysis for MN 133 is here -- https://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/23.7-Sappurisa-S-m113-piya1.pdf -- (at least for the moment, the URLs on that site aren't very stable).
Spiritual practice is not based on dhyana
21a (19) CONCEIT ON ACCOUNT OF DHYANA ATTAINMENT. Furthermore, bhikshus, a false person,
quite secluded from sensual pleasures, secluded from unwholesome mental states, attains and dwells in
the first dhyana, accompanied by initial application and sustained application, accompanied by zest and
happiness, born of solitude.
66 He reflects thus:
‘I am an attainer of the first dhyana attainment, but these other monks are not attainers of the first
dhyana attainment.’
So he praises himself and belittles others on account of his being an attainer of the first dhyana attainment.
This, bhikshus, is the nature of a false person.
21b But a true individual, bhikshus, reflects thus:
‘The Blessed One has spoken of the non-identification with the first dhyana attainment, too. For, in
whatever they conceive, it turns out to be something else.’
So, keeping at heart [keeping in mind] only the practice of non-identification with the first dhyana
attainment, he neither praises himself nor belittles others.
I think that suggests that ...
- The whole sutta is is a list of various types of conceit or objects of conceit
- For each possible type conceit there's a (different) remedy or counter-acting thought, practiced by the true individual
- The remedy for being conceited about attainment of the first jhana is atammayata
... and therefore, again, this sutta mentions atammayata in the specific context of the first jhana.