Obedient: Follows you
Independent: Will move on its own

Opo-opo and Bluebird will get on your Shoulder if non-lala, Head if lalafell if /beckon'ed
Sun will move up/down if /poke'd
'Special' minions will gather, such as Bahamut/Veterans rewards/Beast tribes, or ones that are the same type, such as Mammets/Chocobo's
Dog and Coeurl will gather with each other and fight
Tonberry will stab at whatever enemy you are attacking

May be more, but cant think of them.