IMHO, the reason for the rules being all over the place re: poop is as follows:
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It’s always better training for the horse to keep moving off the rider’s leg and to poop while moving at all three gaits, with as little slowing down as possible.
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It’s always easier to clean up after a horse that stands still and poops.
So facilities come up with all of these weird compromise rules because there is no way to have both–“the horse must stop in the indoor, where the footing is expensive, not in the outdoor, where it’s dirt,” or “no stopping during lessons, only when you’re riding on your own,” or “you must keep going but avoid the poop area for the rest of your ride,” or “instructor will yell at poopy horses she doesn’t like, but ignoring pooping from horses she does.”
I only ask that instructors make it clear to newbies that horses can poop and move and it’s not cruel to do so, because I once rode with a lady who would immediately stop whenever her horse had to go, and when she saw I was continuing to urge mine through the urge (successfully) she told me she absolutely appalled about how cruel and unhealthy I was to make my horse trot and poop.