If I'm not mistaken, only a handful exist. NGC only ever graded one! This one below is not graded by NGC. And, the medium reliefs are described as being a modified high relief. https://auctions.stacksbowers.com/lots/view/1-1N8PO
Thickness at the edge is usually a function of striking pressure. The higher the pressure the better the strike and the more the rims fill on the coin resulting in a greater thickness. For the high relief 21 peace dollars they had to use a higher pressure to try and fill that relief. That resulted in thicker edges but much shorter die life. But the thick edge is not what makes the coin a high relief.
I have problems telling the high relief 22 from the medium relief. They both use the same hub design. I believe the only real difference is the penetration of the point of the bust into the 9 is slightly different. If you had them in hand you might be able to see the difference in relief, but I can't see it in photographs. They are both clearly different from the low relief 22.
The tiara spike next to E is shorter on the medium relief than the high relief. There are probably more little details like that.
Not that I can see. It is long on the low relief and short on both the medium and high relief. I don't see any length difference between the medium and high relief.