I've been sailing for 5 years now and in all that time I've only managed a full planing gybe twice.
That was until today.
I was forever going into the gybe fine but losing it when I switched feet. Getting around but not planing. What I tried today was instead of placing my back foot across the board I put it with the toes facing down the board instead. What a difference
Because the upper body naturally aligns with the feet the sail spun around with my upper body when I turned to face where I was heading and when it went clew first was caught by the wind and spun to the new heading no problems. A couple of times I even picked up a little speed in the clew first position when in the past the force of the wind just pulled it out of my hands. Previously I always found that the body position was awkward and just didn't feel right.
I'm not saying what I tried today is "right" but it worked for me. Give it a go you may be surprised.

Which way is "down"? Toward the nose? over the edge of the rail..?

Sounds like you are holding on to the sail too long - to make a planing gybe, you shouldn't hold the sail onto the new tack until it is "caught by the wind"; doing so means that you will need to really pull hard to sheet in after the sail flip, and as result, you will probably stall the board.

Rather, you should release earlier so that the sail stays neutral, which makes sheeting in the sail (on the new tack) significantly easier -> allowing you to get the power on early, as so not stalling.

That said, there is a definitely a place for clew-first-exit gybes - but not usually for learning how to plane out of of them. Clew-first is good when wave riding or when it is really choppy, as board stability is important.

The real trick to learning the planing gybes, is to find the smoothest water available, eg: on the face of a wave or the inside of a reef, etc. It will allow you to concentrate on when to shift your weight and sail flip - without having to learn to compensate for water-state. Once mastered, then do it on choppy conditions.