Amateur / student here. Sometimes the hardest thing is the first bit of latching onto a new movement pattern - the part that leads up to the moment when you suddenly "get it." Of course that moment is only the beginning of the refinement you are going to do with that movement for the rest of your skiing life, but I'd argue that it's nevertheless the most elusive and important moment. If you don't manage to get there, nothing else matters. And if you do get there, everything else is tweaking.
With this in mind, it seems to me that having a variety of analogous physical life experiences stored away and available to present to the student would be valuable for a teacher, with the hope that ONE of those experiences, at least, will speak to the particular student and maybe help to precipitate one of these moments.
One of these analogs that I've personally found helpful, and that seems to have resonated with some of my skiing buddies who also are cyclists, is as follows. Obviously the exact movements are different between biking and skiing. Therefore from a skiing perspective this is far from being a step-by-step recipe; it's the sensation memory that counts here.
Anyway, there is a way to make a really satisfying, speed-preserving, radius-decreasing turn on a bike that will be old hat to many experienced riders. The best place to practice this is on a slight downhill where you can obtain and preserve speed without too much effort - say, 20mph on a road bike or 10mph on singletrack. You'll also need a defined spot to turn at least 90 degrees, such as a street corner or a sharp turn in the trail. Obviously you need to pick a spot with little or no traffic so you can concentrate on your turn without worrying about hitting somebody! In this example I'll use a right turn.
Get up some speed and coast through the corner in your "regular" way, whatever that is. Cut it as tight as you can. If it's easy to make the corner, you're either going too slow or the corner is not tight enough or both. If you had to hit the brakes so as not to go into the woods on the far side, you were going too fast or the corner is too tight. Either way, try it again until you have a corner that you can barely make without hitting the brakes. Stay off the brake levers completely for this experiment.
Now try it again, but this time make sure your outside (left) pedal is down ("6:00" position), and as you enter the corner move your inside (right) knee in toward the center of the circle and down toward the ground. Move it more than you think you should. It should look "stupid" if you're doing it right. Yes, you absolutely can do this even when you're clipped in. A little better, right? But not a radical difference. Brownie points if you noticed that your upper body wanted to assume a more vertical posture as you did this, to compensate for the additional mass you're moving to the inside of the turn.
Now try it a third time. This time, keep that inside knee action, but push your inside (right) hand smoothly forward as you hit the belly of the turn. That's correct: with this push you are essentially turning the bar left - "the wrong way" with respect to the corner. This is called counter-steering. (Hmm. Starting to ring any bells here, skiers?) What happened this time? If you really committed to it and didn't chicken out, your turn radius was WAY tighter! You can now take that corner faster than you ever thought you could. More brownie points if you noticed that your head and shoulders were pointing in a direction significantly more "straight ahead" than the direction your bike was going at the apex of the turn. THAT sensation is the one that I feel when I get the active inside thing going well in a carved turn on skis.