Q: Is "cocking" the wrist side-to-side motion? A: No.
It is the up/down motion. With the club gripped at address, you should raise the club up towards your right ear using just your hands. Then bring the club back down to the ground. This is the proper cocking and releasing of the hands.

Q: Do I release this position at impact, or maintain it through impact? A: Release at impact.
If you didn't, you would never get the club back to the ground to hit the ball!

Q: Is this up-and-down motion what is to be released at impact? And is this motion like "casting" a fishing line? A: It is what is release at impact, yes. Is it like casting a fishing line?... I don't like to give it that description because it can be constrewed as when to release the hands, like at the top when you really release the hands to cast the line, but in golf you want to release the hands at the bottom of the swing at the ball. Using the hands too earrly will result in a slice becuase the club will be pushed outside the swing path and you will have to pull it back in to hit the ball causing the OUT-to-IN path putting a clockwise spin on the ball which will slice it (right-handers of corse).

Glad you like the site. I should probably put a picture of the hands at a fully cocked position...that might help.