I think he was referring to the list at the top (not in the screenshot) of classes. Only Coercer, Conjurer, Illusionist, Necromancer and Warlock are listed.
Mage is an archetype; Priest, Fighter and Scout are the other archetypes. Sorceror, Summoner and Enchanter are the Mage classes. Wizard is a subclass of Sorceror; Warlock is the other Sorceror subclass. The Mage archetype includes the Wizard subclass.