We normally call the push pull stuff a V, not to be confused with a push-me pull-you. Big V sometimes called a "deep V" or little V. Or a push pull type stick movement with just the wrist, where you move the stick back and forth over the puck without actually changing the direction of the puck would be a "tick-tack" or some pronounce "ticky-tack"
So a tick-tack then deek would be where you move your stick so the back side is in contact then back to freeze the defender then swerve around.

I like 'fade' for the right hander moving to the right side. We normally just say left deek / right deek, but it really is not the same movement. And 'deek' is normally all encompassing for that type of move, whether it is done with a deep V where the puck is brought down under the body before the "swerve" motion, or just the last second "swerve" motion that moves the puck the minimum distance to clear the defender.

What about some team terms? The initial strike? or moves off of the strike? Some locally call it a rush, are there other names for it out there? Do you have names for the set plays off of that?

Advantage pucks? kinda funny aside on that Francios Baton spends some time with our club (being in the Californa wine country) and we were talking about advantage pucks and he didn't know the english for what he was trying to say and just called it a free kick. Talking more many of the terms the french use are borrowed from Soccer, they don't say 'freee kick' that was just the closest translation he could come up with.