STA does not have a consistent formulation for what the Five Ways conclude to:
1.) “this thing all understand as deus.”
2.) “which all name deus”
3.) “which all call deus”
4+5) “and this thing we call deus”
Taking the formulae as equivalent, the proofs all involve some intentional action (understanding, naming, speaking of) of all persons.
1.) All persons. The “all” can’t be understood as what every single individual called deus, so we have to take it collectively, though it’s not clear what collective STA might be targeting. He could mean “all who believe in deus” and so be speaking of pagans, Muslims, and Christians. Read in this way, the “we” is a generic reference to the collective of populations. But we can also read the “we” as meaning “those who do what I am doing now, namely rational theology.” Taken in this sense, the “all” should be read as speaking of a rough consensus among natural theologians. St. Thomas is clearly speaking to a collective consensus, but it is unclear if it is what Aristotle would call a consensus of all or a consensus of the wise. A moderate answer would probably be that he’s targeting all theists or at least a consensus of natural theologians.
The most significant consequence is that, however one takes it, STA is not considering deus as he is spoken of in a any particular tradition. He is not trying to prove the occasionalist God of Islam, the loving God of Christianity, the powerful but fickle local gods of Olympus; nor is he proving any particular variant of deus that one can find in rational theology. He is not excluding all of these accounts of diety, or at least not excluding them all in every way, but he is not providing enough information about deus to flesh out which diety in particular he takes as the true one.
2.) The intentional action. If STA is talking about all theists, then the intentional word is a way of recognizing that the use of the word deus can involve a good deal of error. STA insists that what the Pagans call deus is used analogously to the Christian use of the same name, and so if he wants an account that will include the beliefs of Pagans he has to talk not about the thing itself he proves the existence of, but only the use of the name.
Another reason for this intentional description of deus is that proof for the existence of something usually has to start from the name of the thing you are trying to prove the existence of. Mere naming does not require the existence of the thing named and so serves as a neutral ground to approach the question of real existence.
3.) deus. English distinguishes between “God” and “a god” by use of a capital letter, and then proceeds to edit even Latin liturgical texts in accord with the distinction. STA clearly understood both senses of the term, but his deus did not require him to take up one in the exclusion of the other. What we now call the capital-G god is a being whose existence could never be proven by a single argument but only by a very large and developed treatise, especially if one wanted to speak of a God understood by a consensus of contemporary theologians and athelogians. All one can hope to establish by a single proof – especially a cosmological proof – is a god-like being or a least possible divinity. If, for example, one takes any given cosmological argument in isolation there is no way to avoid Humian critiques a la “for all we know, this designer might be just be a clever or ingenious being, and not an omniscient deity” or “for all we know, there might be billions of first movers and not one supreme God of all gods.” But to think that any of these are meaningful critiques of the Five Ways involves conflating deus with the capital-g god.