As is indicated in the comments and attached chat. I am not familiar with the distinction between man's will being "necessarily moved" and "coerced", but it would seem to arise from John Calvin wishing to retain both the idea that man is a free moral agent and totally depraved, so the man must be "necessarily moved" by irresistible grace to accept baptism and become regenerate, but not be coerced.
The idea that man is a free moral agent and the chief cause of evil (i.e. - God cannot be the cause or creator of evil) is a foundational Christian concept that I trust is universally accepted. So the question becomes where does total depravity come from, since this idea of the will being "necessarily moved" logically flows out of it. Thankfully John Calvin is rather explicit in this regard:
Institutes of the Christian Religion, Book 2 Chapter 2, Part 4
Moreover, although the Greek fathers, above others, and especially Chrysostom, have exceeded due bonds in extolling the powers of the human will, yet all ancient theologians, with the exception of Augustine, are so confused, vacillating, and contradictory on this subject, that no certainty can be obtained from their writings ... still the expression was often on their lips, that man's natural gifts were corrupted, and his supernatural taken away. Of the thing implied by these words, however, scarcely one in a hundred had any distinct idea ... Persons professing to be the disciples of Christ have spoken too much like the philosophers on this subject. As if human nature were still in its integrity, the term free will has always been in use among the Latins, while the Greeks were not ashamed to use a still more presumptuous term ... (autexousion), as if man had still full power in himself.
So John Calvin indicates that his teachings on free-will are, in general, in line with Augustine, but outside of the larger tradition of the Greek fathers and other Western writers.
He practically identifies his doctrine of total depravity (which necessitates irresistible grace) with Christianity itself, shaming those who teach otherwise as being under the influences of unchristian philosophy; which is to say that he thinks the entire Church was confused on a core Christian doctrine, besides Augustine, until he and Luther arrived.
However, its obvious that Aquinas is not referring to John Calvin in his writing. See my discussion with Nathaniel; I would assume with him that Aquinas is referring to some Augustinian interpretive tradition that, like John Calvin, sought to retain the traditional Christian concept that man creates evil while insisting that his nature is corrupted to the point that he cannot choose salvation.
So I suppose this is a partial answer. And thanks Nathaniel for helping me understand some nuances of the topic.