Mini-mash = partial mash. Same exact thing. It's just mashing some of the grain, and using extract for some of the fermentables.
Mini is kind of a misnomber- it can be as big as you'd like. If you have room for 6 pounds of grain, then you'd need less extract. If you only have enough room for 2 pounds of grain, then you'd need more extract. I've made some excellent beers with partial mashing.