The section "A Description of Earthsea" at the end of Tales from Earthsea (2001) is explicit about the hierarchy:
Witchery was restricted to women. All magic practiced by women was called “base craft”, even when it included practices otherwise called “high arts,” such as healing, chanting, changing, etc. Witches were to learn only from one another or from sorcerers. (…)
Sorcery was practiced by men—its only real distinction from witchery. Sorcerers trained one another, and had some knowledge of the True Speech. Sorcery included both base crafts as defined by Halker (finding, mending, dowsing, animal healing, etc.) and some high arts (human healing, chanting, weatherworking). (…)
A wizard, as Halkel [the first archmage] defined the term, was a man who received his staff from a teacher, himself a wizard, who had taken special responsibility for his training. It was usually the Archmage who gave a student his staff and made him a wizard. (…)
Mage remained an essentially undefined term: a wizard of great power.
The story "The Finder", also in Tales from Earthsea, is set in the years that lead up to the establishment of the school on Roke and its rules.
The following quote is an example of the informal use of the term "mage":
Highdrake’s mastery of spells and sorcery was not much greater than his pupil’s, but he had clear in his mind the idea of something very much greater, the wholeness of knowledge. And that made him a mage.
(Highdrake was a mage whose "true name" has been lost.)
The story also tells how the roles of men and women with regard to wizardry on Roke get reversed. Thirty years before Otter/Tern/Medra (the finder from the story's title) arrives on Roke, the island had been attacked by pirates. They could do this because one of the male wizards on the island had lowered the island's magic defences, so the pirates could attack it simply by force. The majority of the survivors were women, and they had little trust in men. In addition, women had always been the leaders in the league of the Hand.
After Medra's arrival on the island, Roke begins to attract more people who want to learn wizardry, so more men come to the island. During discussions about the establishment of a school and how it should be run, some of the men want to exclude women from learning wizardry. Ember/Elehal tells Medra,
And they are men, and they make that important beyond anything else. To them, the Old Powers are abominable. And women’s powers’s are suspect, because they suppose them all connected with the Old Powers. As if those Powers were to be controlled or used by any mortal soul! But they put men where we put the world. And so they hold that a true wizard must be a man. And celibate.
So "wizard" is the "official" term, whereas "mage" is less well defined.