I just think of it as a half-flip landing on the picking foot instead of changing feet. And then the push is onto the LF outside edge, continuing in the counterclockwise direction.
The name is not that important.
If I land on the opposite toepick and push to FI edge, I'm set up to do an inside mohawk to repeat the half-flip, do a whole flip from a mohawk entry, or continue with footwork.
If I land on the picking toepick and push to a FO edge, I'm set up to do an outside three turn into a whole flip (as I do in my current program), or continue with footwork, or whatever.
So which toepick you land on will usually depend on what you want to do next.