Maybe it can simply be a matter of preference? Perhaps magic can be channeled in a variety of ways. Maybe there are ancient words that magi first used, then they found a way to channel magic through gestures, and basically, different magi use different methods?
Also, I wanted to talk about the limitations of magic. I always found it weird how in Harry Potter people could just summon like, stew out of nowhere and then conjure up a bowl and spoon too! I'd prefer it if in this world, you can't just summon physical objects out of nowhere. You can however magically create stuff if you have the proper elements and materials to do so. Like maybe a really powerful alteration magi can create a stone gate by using the masonry from a nearby wall or something. What do you guys think about that?
Antonkun, you mentioned elements. I'm thinking that elemental magic (like fireballs or ice spears or what have you) are conjured, so that would fall under the School of Summoning. Unless you want to make it so that you need to be near an element to use it magically, which I think would be too big of a limitation. As for affinities, maybe magi can have stronger connections to different elements, like Ezrain being good with fire.
Oh, and I really want to make it so that extremely powerful magic requires way more mana than a normal body and catalyst can hold, so, in order to call down these powerful spells, different methods are required, like sacrificial rituals and the like. This is only for the most crazy and powerful magic, like the world-wide memory erasing spells that the magisters use when someone's broken their First Oath.