Stubility

I understand that Misogi generally describes meditation and purification practices some of which take place under waterfalls. And I understand that there is a practice known as 1000 cuts. My confusion comes from having a teacher you used to confuse the two, or rather would call 1000 cuts a misogi or misogi.

Is there a connection between Misogi and 1000 cuts that runs deeper than the general claim that 1000 cuts is a kind of purification? I had thought there was a relation between purification and training in this manner, in pushing the body beyond bad habit, using exhaustion to create more efficient movement etc.