This seems to be the 'collection of points' thread.
On H vs HT - if you fully read that section in Moore, he is making a clear distinction between steps in which the heel raises after the other foot has passed (example - the man's left foot preceding a three step) and steps where it raises before the free foot passes (example - step one of a natural, or the man's right foot in a three step).
This isn't so much a rise, as the start of an upswing - it gets the hips ahead of everything else.
The reason waltz turns have rise end of one is that the hips then lead an actual rise of the body along a diagonal path, still before the much delayed passing of the left foot. The left foot really trails, and only as the swing unfolds is it finally lead to point ahead, reaching but never passing the line of the left side of the body.
All of this happens in the same direction carried through from the preceding step three - there is still no change in the direction of progression, nor any rotation of the feet, though of course there is body rotation commencing from the moment we begin to even think about takin step one. By the the time step two is placed the body will be nearing sideways, but it's only after the left foot is placed that both feet begin to turn to the book alignment of backing DC. As the body fully arrives in step three and the feet close, they turn to back LOD, but the body does not really turn any more. So for the forward half of the natural, we have a body turn between step 3 and the very early part of step 2, and we have a foot turn between late in step 2 and early in step 3.
It's also important to realize that the 3/8 turn natural is a very different step from the 1/4 turn natural, which is much closer to a closed change. In the 1/4 turn natural and the closed change, there is a redirection in the right foot during step one to the inside edge - there's a change in direction of progession such that if you started DW, step two would be taken moving to DC. In contrast, in the 3/8 natural if step 1 moves DW, step two moves DW as well. (these directions of movement should not be confused with the orientations of the feet given in the book).