I'm not sure about current rules, but back in the day, adults were certainly showing both Arabs and Quarter Horses that did not stand 14.2. I had a friend who was very successful in both dressage and hunter classes on a tiny Arab mare and there was a retiree at a nearby boarding facility that had been a very successful show horse. I doubt ths little Arab was 13.3.
I do know that there were adults showing animals that were ponies both by genetics and measurement at a recent Welsh Nationals. Our pony was one of many ridden by a small adult. Nancy