I think that one of the great strengths of Tolkien is the way that “magic” is so understated and apears so natural (in context.) It’s not cheap, theatrical, every-day magic like from Grimm or Sheherazade.
Is Aragorn able to heal because of some “magic” that resides within him, or does he just happen to know lots of herb lore, or is it some combination of both? The text is beautifully ambiguous.
Similarly, for instance, when Gandalf is riding into battle, shining in white – I don’t have the text handy, so I can’t quote exactly – like the rays of the sun, and it inspires the troops. Is that “magic” or is it just that Gandalf is very charismatic?
The elven rope that comes untied at just the right moment: is that a magic of the rope, or is that sheer lucky coincidence?
Or the snows that block the Fellowship atop Caradhras. In the movie, it’s clear that the storm is a sending of Saruman. In the books, however, it’s deliciously ambiguous – is it a magic of Saruman, is it just bad luck, is it some malice of the mountain itself? We don’t know, the characters don’t know, it’s up for speculation.
The Ring feels heavier as Frodo moves into Mordor. The lure of the ring is in greed, we never see actual “magic” (aside from the invisibility trick.) Invisibility is a minor trick the Ring can do, but the main magic of the Ring has to do with lust, greed, and power. It’s not the trite type of magic where rubbing the ring causes a genii to appear, or whatever. Tolkien’s magic is much more subtle and more “natural.”
Even the bits that are clearly “magic”, like the Phial of Galadriel that becomes a light when all others go out – but it’s flamed more by courage than by incantation.
IMHO, the way that Tolkien makes magic seem “natural” is one of the joys of LotR. And that means that you will never resolve this discussion. Even if Tolkien were here, I suspect he wouldn’t answer the question, he would leave the ambiguity.