I do that but I also whisper softly in it's ear, "Duck Season, Wabbit Season" when I make the cut.Yeah, this. Do the knife thing, and make sure and come back and post what happened.
I think the general rule of thumb is you do what preserves the most meat. Crush the skull, pull the head off, twist the neck. If you can't stomach pulling the head off by hand, step on it and pull. I am talking small game and birds here.
As for large game, approach from behind so it runs AWAY from you. Long stick to poke the eye. I keep my muzzle back away from the Deer where I have control of it and it won't get hit, kicked or pushed aside.
PS I never cut a deers throat to finish it.