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As far as "types", we don't have much in the way of clear distinctions.
There's obviously the Mid vs. Belka distinction. There's certain characteristics of each (Belka's combat-oriented, to be sure), but we don't ever hear that this or that is "impossible" for either of the two systems, so there's a lot of overlap. They have, for example, very similar barriers. The two aren't mutually exclusive - there's "Neo Belkan", which incorporates a lot of Mid, and Hayate has a variety of both types of magic from the Book. Unique to the Belkan system is the Unison Device, though we don't know precisely if it can ONLY be Belkan or whether that's just all we've seen.
We only know a couple of summoners. It seems like a different class of magic, but Lutecia is still capable of offensive action, even if Caro can't. The kind of augmentation spells that Caro uses aren't unique to summoners either - Sonic Move, for example.
The Inherent Skills that are used by combat cyborgs are a completely different class of magic. Much less flexibility, of course. Nor does the possession of an IS mean you can't use other magic (Subaru).
Other than that, who knows? There could be a bunch of other types of magic, all lumped in with "Lost Logia". It's also possible that other completely different systems of magic could be generated - clearly Mid and Belkan styles show that there are multiple methods of producing similar effects.
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