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I finally got a handle on the meaning of the expression, "omne quod movetur ab alio movetur"

That is the Latin expression used by Aquinas in his first way of proving the existence of God.  It means all that is moved is moved by another.  All it means is that things that are moved are not moved by themselves but are instead moved by another.  It does NOT mean that all things that are in motion are moved by another.  The latter claim actually contradicts what Aristotle and Aquinas thought about natural motion (up/down for light/heavy things).  That is, heavy/light substances naturally move down/up in virtue of their form.  No need to posit an efficient (moving) cause.  Not unlike what we would say now about gravity, the four forces (or is it five this week?).

 [digression: the real point of Aristotle in saying {in Greek, not Latin, of course} omne quod movetur... is to contradict Plato's notion of a self-moved mover: no such thing possible for Aristotle]

All that stuff people say about the need for a medium is inapplicable to natural movement.  See Aquinas's commentary on De Caelo, III, lec. 7, par. 9: no need for medium of natural movement.  So all that talk in the middle ages about the need for a medium even for natural movement comes from Averroes, even though it is attributed to Aquinas (sez Weisheipl in "Omen quod movetur" article in his book on Nature and Motion in the Middle Ages).

But when it came to circular and violent movement, Aquinas had different opinions... ones that we would find hard to swallow.  Circular movement is not natural by definition because rest, not motion is natural.  But it is not violent either.  It is caused, in Aquinas's opinion, by intelligence.  Either the spheres are animated by souls or moved by angels.  Take your pick.

And when it comes to violent motion, there must be a kind of struggle... and THIS is where a medium comes into play.  Throw a stone up and the air supplies a force that either keeps the thing moving against its own natural inclination or it supplies a kind of object of resistance that enables the projectile itself to struggle (temporarily) against its own natural inclination.
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