The mage gains the incapacitated trait while stunned. All the other affects from incapacitated apply but the only differencee for mages is that they can still cast quick, non attack spells with it. So no moving, no attacks that's aren't independent of the mage (like dancing scimitar), and no full casting (creatures or destroy magic, crumble or disperse and the like). So your friend is correct. You cannot take any actions at all while stunned as a creature, but a mage can really concentrate hard and get a quick spell out, as long as it's not an attack.