Is “man-at-arms” just a generic term for a soldier who isn’t a knight, roughly equivalent to an enlisted man rather than an officer, or is it more specific than that?

It’s a bit more specific. A man-at-arms is a professional soldier who has been trained to the level we associate with knighthood, i.e, learning how to fight in full plate with a variety of weaponry (sword, axe/mace, polearms, lance, etc.) both on foot, but especially on horseback, and who owns the arms, armor, and horse necessary to fulfill that role. 

So by definition, all knights were men-at-arms, but most men-at-arms were not knights. Most men-at-arms came out of the gentry or esquiry, because pay for men-at-arms was quite high and the possibility of social advancement into the lower nobility was good.