In my opinion, what makes a mage is largely up to the writer.
Personally, I try to define mages by the source and flow of magic. For example. I wouldn't say that a street magician performing tricks using slight of hand is a mage, because there's no magic there, just dexterity and skill. A Druid, someone who has learned to channel the magic inherently found in nature would be a mage, because they learned how to apply pre-existing magic. A scholar who finds a forgotten tome that gives them dominion over a particular thread of magic would be a mage.
If you really want to go in depth, try to think about how the magic works. Such as Alchemy. Alchemy ala The Elder Scrolls is essentially just science- mixing ingredients to make compounds that have various effects. Whereas Alchemy ala Full Metal Alchemist is magic- imposing one's will over matter in order to alter is.
Again, it's up to you, as the writer, to decide what makes your character a mage. Just remember that your ideology and the ideology of your characters can and likely will differ from the ideology of other writers and characters, which is okay! Character development can come from that~
Hope this helps!