So in my works that use magic, magic functions similar to how you're describing... all kinds of systems are functional and there are few hard rules to it. It's been my experience that magic functions in one of two ways: Magic is Physics/Math... it has lots of rules and procedures developed over years of explortation. That which is possible is and that which is not possible is never possible. Your system is the other type, Magic is Biology: There are rules that work, except there is almost always some kind of exception to that rule (the one of my favorites is that all animal cells do not have Chloroplasts... except Cyanobacteria, which is classified (or was) as an animal, but primary enegy source was Chlorophyll).
In the former system, there are books that describe precise form, movement, formula, and technique to invoke the spell... and if done correctly, the spell happens. In the latter, the technique and formual are not important, but the mindset is. Essentially, for this one, it's teaching "think outside the box".
In my own personal system, Magic is derived from systems like Mathmatical Proofs and legal loopholes, rather than a set spell that works. This is because Magic is the powerful belief that the world works on a certain logic. It actually, comes out in planning at some point, that the hardests of sciences are also magic in and of themselves... Physics, Chemistry, and Math are the most commonly accepted maths... many a good wizard are actually the worlds best scientists... they just don't call it magic.
Essentially, what would be magic in this system is the ability to "Reject your Reality and substitute your own." I answered a similar question yesterday about this, but I use the example of making a free-standing door that is in the middle of a field can open to a room unseen, even if you can completely walk around the door. Here, the wizard relied on the nature of what a door is: A passage through an otherwise impassible barrier, and works out that just because there is no barrier that he can precieve with his senses, it does not mean that the door is not passing through a barrier that exists beyond his senses. Either there is an illusion, or the door opens in a direction that is not 4 Dimensional in space-time. Thus, by accepting that the door must always function as an opening in a barrier, he can use it to open to a barrier he cannot percieve through normal senses. Meanwhile, your muggle is a practitionare of physics and so entrenched in that system, that the thought of a door fuctioning that way is absurd and the door will never work like that for him.
Similarly, in this system, breaking a spell relies on understanding the logic of the Mage behind it and exploiting that logic to find a loophole. In fact, a truly logical magic is impossible because the whole system operates on magic logic... thus, if a mage can "prove" his spell is possible, the quickest counter is to "prove" a condition that is impossible. Simpler spells require less of a leap in logic than complex spells, and thus are a bit harder to counter directly, but do less damage. A complex spell might be highly dangerous to those affected, but also are filled with holes that are just waiting to be abused.
This makes the system less about genetics and destiny and more about mindset and imagination. Anyone can be a Mage does not mean it's easy to learn this mindset. But it also doesn't limit some of the best Wizards having science backgrounds.
Teaching such a system would look more like a law school: Law schools don't teach laws, which are subject to change by legislatures and judges, but law theory and logic. After you get that degree and pass the bar, your employer will teach you more practical aplications (such as court room procedures and laws related to their practice). Like your magic system, there are few rules that can't be loopholed if you have a good mage/lawyer... but what makes a mage/lawyer isn't so much knowing the system, but knowing the system well enough to get away with it. It's been said "Good Lawyers know the Law. Great Lawyers know the Judge". Any lawyer can file a motion to dismiss evidence... but the lawyer that figures out the judge trying the case is more receptive to the defendant after his lunch recess is going to get paid better because he can be more successful. It's not illegal... it's barely even an ethical concern... but a happy/grumpy judge is not something that many lay people are aware of.
This still allows for big tomes of magical knowledge. Merlin might have been a great Wizard and came up with many powerful spells, but to truly be powerful when dealing with Merlin's spells, you need to understand the way Merlin thought. It's not so much here is how Merlin's spell works but here's WHY Merlin's spell works.
Some courses to consider:
- Logic (You cannot reject logic if you do not understand how it operates) and Magic Logic (may take the form of "Wax on Wax off, Daniel-San" and "Try not. Do or Do Not, there is no Try" and "There is No Spoon" sage like non-sense babble... it's primary purpose is to teach people how to unlearn things.).
- Spells, Basic, Intermediate, Advanced (teaches some common spells, their logics, their flaws, and their counters. Higher classes up the complexities and the ability to integrate two logicals that should not work together).
- Geometry (mostly taught because it's proof heavy).
- "Traditional" Magic School courses. A Defense against the Dark Arts might deal with certain malicious theories. If magical flora and fauna exist in your world, they must have their own magic logic too them... since they are not nessearily self aware, these logics might be as hard for the life form to reject.). Transfiguration would focus on accepting the fuidity of the phsycial form (if you're rejecting physics, than that means that which is physical is not subject to pesky things like conservation of matter/energy but keeping the mind intact is difficult.).
- Specialization would probably take the form of an apprenticeship to an accomplished master... these could probably take the form of the Master first recognizing the pupils propensity for creative logic that is like the Master's own, sponsering his pupil's classroom education, then apprenticing the pupil in order to teach the pupil his own school of magic and allowing the pupil to develop a unique magic unto the pupil.
So I didn't mean this to act as a counter to your system. I agree it works well with yours. My point is in writing a magic system, there are one of two ways. The physics one describes something like Harry Potter, where magic works because of a defined system of rules. Understanding how magic works is not necessary to the story as it's about using tools for good and evil purposes (or more right and easy as conflicting ideas).
Your system I would liken more to Avatar: The Last Airbender. What makes a bender verses a non-bender is never explained, but benders are clearly shown to need certain mindsets to use their skills. No one style is shown to be superior and the series puts great emphasis on both fundamental perception of how a skill works, greater understanding of your own school allowing for more specialized skills (the sub-bending styles), and understanding different schools allowing out of the box thinking to develop new skills. For example, the user is shown that Waterbending incorperates water as either ice or liquid water. So when we're introduced to the idea of pulling warter from the air, the viewer can see the logical flow. By showing that waterbending works on pushing and pulling the water, we can later accept that it can be pushed and pulled from a closed system, such as a vine, which allows us to accept plantbending. And by accepting that water can be bent without being seen inside of a closed system AND that water can be used to heal the body, we're able to accept the concept of Bloodbending. Likewise, we are given logical progression of "Earthbending was developed by mimicking blind badger-moles" to "An Earthbender can use the earth to see by the vibrations" to "An Earthbender who by vibrations in solid materials can see impurities in metal, allowing her to accomplish the previously impossible techinique of bending metal." There is also examples where Aang has to master a techinque by defying a much more ingrained idea, and Korra incorperating a new concept in order to master an old concept she had difficulty understanding. In order to understand these accomplishments and why they are significant, you have to understand how the system works to some degree of physics. The show has demonstrated numerous times that no one bending school is inherintly superior (or even lacking in a bending school) but rather that it was the skill, cleverness, and technique of the individuals that won out. As codified in the Episode "Sokka's Master" Sokka wasn't selected to be taught because he knew all about sword fighting like all the other potential pupils that sought the titular master, but rather he was selected because he was able to admit he knew enough to know he knew nothing about it, which proved to the master that Sokka would be more receptive to his teachings specifically because Sokka was aware of his limitations in the field.
For this style of magic, again, it's not about memorizing and regurgitating a spell, but teaching mindsets that can make you open to the possibilties of magic. Your lessons should give the pupil, and thus audience, an insite into a new way of thinking.