I can't easily think of more, though I can think of some variants or combinations of two options.One common idea is that there WAS a language of magic, and various ancient languages are a corruption of it. Hence, many spells being in latin.Another, which I've not heard spelled out, but is my interpretation of how it often works implicitly, is something like this. A spell is found by some unspecified sort of magical research, combining together various elements including a verbal incantation. In principle, a similar effect could be achieved in a completely different way. But the way found by research and trial and error, experimented with various verbal components, and tried faux-latin ones first, because we have some vague generalisations about which words might be useful from their use in other spells, and moved on to other languages, meaningless sounds, etc later, all the time experimenting with other components. That explains why most spells are faux-latin, but some are something else. If we had a completely different language, we would have discovered different combinations of ingredients to make different spells.Although that's kind of an excuse, enough to justify the world-building, but without any real predictive power.
there WAS a language of magic, and various ancient languages are a corruption of itThis would be particularly amusing in the Potterverse for the implication that the bad Latin grammar in the spell incantations is actually right, and the thing we think of as 'correct Latin' is the wrong, corrupted, half-arsed version!(And, just as when I half-remember a tune my brain will make up a plausible version of the missing parts that then turns out not to match the original, the Romans constructed a basically consistent grammar around their corrupted bits and pieces of spell-speech, it just wasn't the same as the one they were trying to recall.)