Ninjas combo finishers are all a bit, uh...confusing.

Huton is a wind elemental buff.

Aeolian Edge comes from Aeolus, ruler over the winds. To match, it has a green-ish hue that indicates the wind element and has you spiralling high into the air.
No interaction with Huton whatsoever, only does damage.

Armor Crush is a simple piercing attack much akin to trick and sneak attack.
Increases Huton duration.

Dancing Edge has you wildly flailing at the enemy. And, somehow, reduce slashing resistance in the process.

Like...wut?