What is a king vessel? Is it a dungeon capturer that is rewarded with a djinn? Then isn't it no difference from a metal vessel user?
A King Vessel is a dungeon conqueror who was chosen by the djinn of the dungeon and rewarded with a metal vessel containing the power of the djinn. Almost always they are chosen by a Magi who helps them through the dungeon.
A member of the Kjng's household is someone loyal to them who carries a household vessel, imbued with power by the same djinn the King controls, they have not themselves conquered a dungeon, and have not been chosen by a Magi. They do not have the full power of the djinn, and are unable to use Extreme Magic.