So, for anyone wondering how to fit in your ideas from the original system to this new one, I'll translate the arts below:
Universal Magic now covers effects previously labeled under Cerebromancy.
Air Magic now covers effects previously labeled under Aeromancy, Electromancy, and Hydromancy (moisture droplets in the air/humidity).
Earth Magic now covers effects previously labeled under Geomancy and Ecomancy.
Fire Magic now covers effects previously labeled under Pyromancy.
Water Magic now covers effects previously labeled under Cryomancy and Hydromancy (larger quantities of water than used in Air Magic).
Infernal Magic now covers effects previously labeled under Necromancy, Hemomancy, and Osteomancy.
Cosmic Magic now covers effects previously labeled under Chronomancy and Psychomancy.
With this new system, we'll say some Magi learn basic Universal Magic from a young age, which covers any effects not tied to one of the four Elemental Planes, the Infernal Abyss, or the Cosmic Void. This can also cover minor healing abilities for mild surface injuries, cuts, bruises, burns, etc. Kind of like the magical equivalent of having an at-home first aid kit. For greater healing, people usually rely on Alchemy, which includes potions, herbology, and transmutation. Using that, they can brew healing elixirs and transmute damaged flesh back into healthy flesh. This would technically also fall under Universal Magic.
That being said, healing effects can also occur from using more advanced elemental affinities:
Air Magic can generate a healing water vapor that is inhaled and boost's one's own natural regenerative process.
Earth Magic can generate a healing bioluminescent light on the caster's palms that draws out or mimics the properties found within herbs used for regenerative potions.
Fire Magic can generate a healing flame that momentarily melts injured flesh and then restores it to a healthy state. However, this can be horribly painful if not done properly, so it's not often done outside a controlled medical setting or emergency situations.
Water Magic can generate a temporary liquid cast around an injured body part, drawing out the natural healing properties of the water and directing it to the wounded flesh.