Sorry - the grip I described was just my starting point.
I begin by reaching up and establishing the top-side grip I described, then I lift my feet so I'm hanging below the rings, and I start the pullup part. By the time I've pulled up to the transition point, my sweaty hands have usually slid down into a position that lets me move through the transition and push up out of the dip.
Does that make more sense now?
Does it sound kosher?