Whereas to both of you, the defining characteristic of magic appears to be the pacing mechanic?
"Pacing mechanic" undersells what it is. It's not a stylistic choice, it's not about timing how long your session is going to run or how much a player can do within one, it's a description of how much [mana/stamina/mojo/etc.] your character has, and a description of how long he has until he becomes unable to do his thing. It's not trivial.
Moreover, in the specific context of D&D, the Vancian mechanic is defining. In another rpg, it might not be, but D&D magic generally has a per-day component. Nonmagical things generally do not have daily limitations in D&D (no, hit points and healing are not a daily mechanic), which is why the exceptions (i.e. Bo9S, the PHBII knight, or all of 4e) are referred to as spells, again specifically in a D&D context.
It's also worth noting that in the specific case in point, the mechanic doesn't represent any kind of fatigue; you might "run out" of one power but have several others, and even if you run out of all of them, your character is not tired at all, simply out of powers. This makes no sense. For magic, we tend to make allowances for things that make no sense, which is why Vancian magic is accepted. Thus, the other thing that makes powers spells is that there's no explanation for why they work at the intervals they do; at least, not a satisfactory one.
Lastly, consider why the power system was conceived in the first place: to try to balance characters with spells against characters without spells. The underlying intent was to make martial exploits and spells balanced, and the same supernatural-sounding terminology (powers) is used to describe both. So this equivalency is not something that some ENWorlders made up out of nowhere; it really derives naturally from what the 4e designers wrote.
None of this is to say that your other definition...
It is that they easily and overtly break the laws of physics (I sometimes think some of the Olympians are breaking them not terribly subtly right now).
...is invalid. This is absolutely valid. Magic can have multiple characteristics.