Faliry
1. (Mechanical Magic): it enables its user to do all sorts of things you usually can do with your hands or tools (move, throw, cut, punch holes, sew, slice, assemble, etc) with a simple spell.
These could be aspects of Transmutation or if they're not actually change the object but using a force like you would use a saw, then it could be some form of telekinetics (or maybe just kinetics).
Faliry
2. (Elemental Magic): its user can produce and control fire, water, ice, lighting, etc. (kind of like RPGs' black magic)
You could refer to it as Evocation, each one could be it's own school, or you could remove it entirely and refer to it as results of other magics. For example with using telekinetics you could vibrate molecules causing the air to burst into flames.
Faliry
3. (Medical Magic or Physical Magic): just like white magic, its user can heal wounds, relieve pain, heal some magical conditions and give some buffs, like increased agility, strength, stamina, etc.
This could also be aspects of Transmutation. It could also be Enchantment, especially if they're giving people properties they don't normally have. There is also Necromancy, at least with the more modern definition, which reworks flesh (usually dead flesh).
Faliry
4. (Physical Magic or Martial Magic): kind of like monks, this kind of magic allows its user to control a magical energy inside their body to increase its strength, reflexes, endurance and such and even heal their own health (it can't be cast in someone else though).
Definitely Enchantment, abet self inflicted.
Faliry
5. (Psychic Magic or Mental Magic): enable its user to inject emotions, feelings and thoughts in another person, subjugate others will and things like that.
Clairvoyance would allow to see into the minds of others, half way across the world, or hours/days/years into the future (or past). Illusion could cause people to see and hear what is not there or even be able to touch it if it's strong enough. It could also be used to put thoughts into another's mind.
Faliry
6. (Natural Magic): a little like blue magic, its user is able to control plants, animals and monsters.
Shamanism if you're not so much controlling as persuading the plants and animals to do tasks for you. If you're truly controlling the animals this could be a form of illusion.
Faliry
7. (Alchemy or Brewlogy): enable its user to imbue magic while merging ingredients to create all kinds of potions and bombs.
Alchemy is alchemy. It can be aided by other types of magic like transmutation but it's still alchemy.
Faliry
8. (Transmogrification): magic that makes possible to change attributes (color, size, shape, texture, etc) of things and even beings.
Transmutation. This is exactly what transmutation is about.
Faliry
9. (Tinkerism): not exactly a magic, but it's like a science that does not obey real science laws, but needs a lot of study, calculus and applied concepts to create machines powered by crystals previously imbued with magick by a mage.
This could be considered an aspect of alchemy. You could also just call it Science, after all if your world support the use of magic then that means magic is apart of the laws of physics and that this type of technology doesn't actually violate the laws of physics but rather utilizes them. Unless it turns out that magic is just energy leaking in from another reality into this reality then you could call it something like Demechanics (demon or dimensional mechanics).
You might want to look up Hermetics, Chaos Magic, and Formulaic Magic (which is the name of a trope of magic involving advance mathematics).