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Having read these questions, I've some further questions as the title says. From those preceding questions, we can see that magic is innate to Arda, and that all sentient beings can use it given adequate and proper study. In other story universes, magic that has to be studied is frequently divided into different "schools", sometimes for purely academic purposes and sometimes because that is simply how the nature of magic is. The texts do suggest that people in Middle Earth do distinguish between different forms of it - Galadriel views "Elf-magic" as being different from the "deceits of the Enemy".

So, with that in mind, do we know if the peoples of Middle Earth understand/study magic in terms of some form of taxonomy into various fields/disciplines/schools/etc.?

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    No. The only thing to distinguish them is that some magic comes from knowledge taught by the good Ainur and some magic comes from knowledge taught by Morgoth and Sauron. (But plenty, it must be assumed, was devised without tutoring.) The difference is in purpose and means, not in kind. – Shamshiel Nov 15 '15 at 13:45
  • @Shamshiel note though that a "school" of magic can also mean an approach, without necessarily being fundamentally different in kind. Think of schools of art, for an analogy. In that sense, there may well be multiple schools of magic among the peoples of Arda – Darael Nov 15 '15 at 20:36
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Not exactly. But there are different kinds of magic, and different philosophical approaches to their use, which depending on your perspective may amount to the same thing. Tolkien discusses this somewhat in the unsent Letter 155:

[F]or the purposes of the tale, some would say that there is a latent distinction such as once was called the distinction between magia and goeteia. Galadriel speaks of the 'deceits of the Enemy'. Well enough, but magia could be, was, held good (per se), and goeteia bad. Neither is, in this tale, good or bad (per se), but only by motive or purpose or use. Both sides use both, but with different motives. The supremely bad motive is (for this tale, since it is specially about it) domination of other 'free' wills. The Enemy's operations are by no means all goetic deceits, but 'magic' that produces real effects in the physical world. But his magia he uses to bulldoze both people and things, and his goeteia to terrify and subjugate. Their magia the Elves and Gandalf use (sparingly): a magia, producing real results (like fire in a wet faggot) for specific beneficent purposes. Their goetic effects are entirely artistic and not intended to deceive: they never deceive Elves (but may deceive or bewilder unaware Men) since the difference is to them as clear as the difference to us between fiction, painting, and sculpture, and 'life'.

The Letters of J.R.R. Tolkien 155: To Naomi Mitchison (draft). September 1954

The key difference isn't what magic is used, but the means it's used to achieve: the "good" side uses magic for beneficent (lighting a fire on a snowy mountain to keep from freezing to death) or artistic (the Silmarils) effect; the "bad" side uses it to dominate the land and people.

However, despite the conclusions made in some answers to the linked questions, the idea of "studying" magic seems fundamentally incorrect; Tolkien notes near the end of Letter 155 that magic isn't a matter of lore, and that it's not attainable by men:

[A] difference in the use of 'magic' in this story is that it is not to be come by by 'lore' or spells; but is in an inherent power not possessed or attainable by Men as such. Aragorn's 'healing' might be regarded as 'magical', or at least a blend of magic with pharmacy and 'hypnotic' processes. But it is (in theory) reported by hobbits who have very little notions of philosophy and science; while A[ragorn] is not a pure 'Man', but at long remove one of the 'children of Luthien'.

The Letters of J.R.R. Tolkien 155: To Naomi Mitchison (draft). September 1954

At best we might say that "studying" magic involves learning to use it for different purposes, as Celebrimbor learned Ringmaking from Sauron.

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