That would seem fine - after all, it wouldn't help the Magus teach the student other arts, or Latin, or Magic Theory, etc - so it still has important limitations,
Generally I would say that was fine, although I think "teaching magical abilities to those with the gift" might be better,
My only reservation is that it is very easy for Magi to never teach anyone without the gift, but then I suppose it is also very easy to never wield anything but a shortsword with your single weapon ability too - so I suppose it isn't totally ridiculous,
You might want to reread the beginning of the Abilities chapter. It doesn't work that way. The list given is not definitive. They are only "suggestions." "Other specializations are possible with the permission of your storyguide." Teaching (Corpus) is fine if your troupe/storyguide is fine with it. I'd say it's a lot less open to abuse than an Ability since you can only instruct one person at a time in an Art. Still, if you'd like to teach Arts well, go with Teaching (Apprentices) or Teaching (Hermetic Magi) or something like that. Those will serve you better.
It seems to me that those specializations are of the same order as Teaching (Academic Abilities) or Teaching (Martial Abilities) and much larger than a particular Ability. Not that I think there's anything bad with that but it's worth having a second look.
I agree. But on the scale of "particular kinds of students," they're fairly narrow. Even as broad as they may seem, I'm not sure they're open to too much abuse. How often will you be teaching other magi? If it's a +1 when you're teaching your apprentices (not so likely to be teaching others), that's probably not too scary. Meanwhile, something like Teaching (Martial Abilities) in the hands of a non-magus focused in combat would probably come into play frequently in off seasons because the magi would want that person constant instructing the grogs and many of the companions.
Being specialized in teaching an Art makes perfect sense to me, although it does seem overly specialized - a specialization in teaching the Arts would probably work.
As for types of students, I'd go with "Apprentices" or "Magi" but not both. This seems better balanced and it would also make sense that someone adept at teaching beginners wouldn't be as skilled with more experienced magi.
But it's deceptive. Teaching (one-on-one) provides a much smaller fractional bonus than named groups do. That's because of the +6 everyone gets for one-on-one anyway. Typically the rest of the bonus will be no more than double that. So we're comparing a +1 out of 12 or so to a +1 out of 18, but in different situations. You also lose out on a different thing: numbers of students. With other group types you can also manage more students, but not with that one since you'll cease to be teaching one-on-one.
With my own teaching-focused character I found Hermetic Magi to be a good specialization. I was also exceedingly good at Magic Theory (Affinity helped). I then ran Magic Theory classes at fairly low cost. Would you pay 2 pawns of vis to get to learn from a +20 source? Since my Magic Theory was high, many magi could study under me repeatedly. Since my Teaching was 7 (w/ specialty) I could handle teaching many, many magi at once. That's what allowed me to keep costs way down. Sure, it cost me lots and lots of silver to feed and house everyone for a season, but I allowed some to substitute some silver for vis at the typical redcap rate. I was able to make sooo much vis in one season it was scary.
While this is true, any teaching a magus will do/require will always be one-on-one. Apprentices have to be taught one-on-one, and Arts can only be taught this way. Even if you want to teach MT, no magus would pay for teaching if it's not one-on-one. The 6XP extra are just too good to give up.
Who needs numbers of students if you're only going to teach magi/apprentices?
It's the quality of the source that counts, so why quibble over the circumstances?
If you have the choice of paying the same cost to go with a class and a quality 20 source (because the teacher doesn't consider one-on-one worth his time - he can charge each student less, but earn more overall working with a class), or individual tuition with a quality 10+6 source (or even a quality 14+6 source), who're you going to go to?
Remember - the teacher has some input in how his time is spent too, and if he considers teaching classes to be more profitable (and possibly cheaper for the students, as well as freeing up more of his time for other things) then one-on-one may simply not be on offer,
Indcidentally, when teaching a class, do all the students need to be "at the same level" - it seems like teaching a class with varied ability levels should be seriously penalised (you don't put the primary schoolers in with the sixth form),
Magi may not be "social animals" (though I suspect that really is an overgeneralisation) but they still mostly live and work in groups (covenants) and get together at major gatherings,
I also suspect that they would be more inclined to be sociable around each other rather than with mundanes where the gift is more of a problem,
Fundamentally, if I can charge 2 vis each for 5 students then for one-on-one I'm going to want 10 vis for the tuition (and maybe more, because I would rather not spend all of my life teaching, so I'd rather get it all over with as fast as possible). Is it really worth the cost?