Attempting to create a Magic System that generalizes all types of magic

Im trying to use all types of magic in story and i am trying to come up with a formula for it (it just came naturally) and I having a tough time calling it.

Magic used by wizards, warlock, witches, arcane users, or elemental magic, and the like from Harry Potter and Avatar to Fairy Tail and other anime have something in common. Magic from Harry Potter use magic (lets call it this for now) "Indirect magic". This means magic users use an object to use their magic like wands, staffs, books, ingredients, swords. While "Direct magic" users basically have magic come out of their bodies themselves, usually by hand. This comes out looking like a fireball coming out directly our of their hands or a magic circle comes first, then the spell.

I plan to use this in my story as something that separates the weak from the powerful because of the different methods in using magic, but I dont have a solid name for it yet. Any suggestions?

The other problem is explaining how it works so the reader can understand them like it was an instruction book while maintaining a sense of wonder.

So far, what Ive come up with looks like this: Magical Energy + Intention = Spell

This works well (sort of) for Direct magic users and I can use it in explaining how to create a new spell.
It gets more complicated (I think) when I try to do this with the Indirect magic users. There are so many types of them but it always need magic energy first then intention, but you can also get rid of the intention part if your a witch that uses ingredients to cast a specific spell. If you want to harm someone, just use energy, then the ingredients needed to cast the spell.

Am I overthinking this (too fleshed out) or not enough? Can you see what else is wrong with this since I am totally blinded right about now.

 

High and low magic huh? I am not really trying to fuse them all im, I just trying to properly differentiate them. What about Arcane and Sorcery? Would that be a good the two different methods of magic?