If one intends to form a country where ones puppet is needed, is it NOT enough to be the liege. The only way is putting it in ones sphere of influence.
Dominions are countries that are your same culture group. If you're Spain and you vassalize Portugal, Portugal becomes a dominion instead of a satellite. However if France vassalizes Portugal, they become a satellite. Dominions are more deeply bound to the liege than Satellites.
In [[HOD]], one can take direct control over ones puppets troops.