Wizards are people that have a talent for magic that's genetic. They live normal life spans like any human. ( Harry Potter kinda)
Sorcerers/Alchemist are a house of wizard that studies magic to explain why and how it works to manipulate it. They turn their physical energy into magic using sorcery to manipulate the laws of nature. This can lead to a shorter lifespan after awhile.
Magi/Mages are rare and pop up randomly. There use to be a lot of them back then, but they declined since the great war(like all the others) and are endangered. They have a way longer lifespan than most. Magi can see the neighbours and use them to help with magic so that they can break laws to get what they want. This strains them if they use magic alot. They can do magic with their will, but it is problematic.
I hope I answered your question and got nothing wrong!