Re-reading the previous post, I think it probably would be handy to lay down explicitly what possibility, impossibility, and necessity are under this conception. This will be crude, but should give the idea. A thing is possible to be in this sense if for some time everything required for its existence exists, and in that way that makes the thing actual; a thing is possible not to be if for some time not everything required for its existence exists. (There are Aristotelian arguments that these two are coextensive.) A thing is impossible if this is not true at any time that what is required for its existence exists. And a thing is necessary if at every time what is required for its existence exists, and in such a way that the thing actually exists. You'll notice that in every case we are dealing with a modality that depends on what actually exists: to be possible to be and not to be is to be such that sometimes it actually exists and sometimes does not; to be impossible is to be such that it never actually exists; and to be necessary is to be such that it actually always exists. The power/ability/capability here is one involving sufficiency; it is always expressed (the ability to be in being and the ability not to be in not being), and what kind of ability it is depends on how it is expressed.
The Third Way, therefore, is an argument that proceeds from the fact that things are generated and corrupted to the fact that they are generable and corruptible (possible to be and not to be); these are understood not in our weak usual sense of 'could be generated' and 'could be corrupted' but in a stronger sense that's more like 'having at some time the sufficient conditions for existing and having at some time the sufficient conditions for not existing'. But if something is generable and corruptible in this sense, it is logically impossible for it always to exist. If the whole world were generable and corruptible, at one time nothing existed. But nothing is caused to begin to exist except by something existing, so if at one time nothing existed, nothing would ever begin to exist and nothing would exist now. Therefore not all things are generable or corruptible: there must be something ingenerable and incorruptible (necessary). That is, there must be something that does not have to begin to exist and does not ever cease to exist. But anything of this sort would have to be such that either it is ingenerable and incorruptible by nature or it is caused to exist in such a way that it does not begin and end, i.e., it is ingenerable and incorruptible at second-hand. Now, this is basically a question about whether efficient causes can regress infinitely, and Aquinas already dealt with that in the Second Way; the causes are considered from a different angle here, but that's all. So there must be something that by nature is ingenerable and incorruptible, which can cause things that are not ingenerable and incorruptible by nature to be ingenerable and incorruptible through its causal activity. And this is the sort of thing human beings call divine.