Limmī: the middle term grounds the judgment and is also the real cause of the major predicate belonging to the minor subject.
Innī: the middle term grounds the judgment but is not the real cause; sometimes it is an effect of the relevant cause.
Both share an epistemic middle term. The defining difference is whether that term also has the relevant causal role in reality.
To classify an argument, identify its conclusion, locate the middle term, and investigate how that term is related to the concluded fact in the world.