Around here we have "English Pleasure" which is similar to hunter under saddle as it mainly focuses on the horse, and "Basic seat" which is similar to hunt seat as it mainly focuses on the rider. The difference being that unlike hunter/hunt classes, you dont have to ride in a forward seat, and is judged more on your all around english horse and flat riding style, as oppose to a horse you're going to see over fences. We get a range of hunters to dressage horses/riders.
For the other one, "Cross rail Equitation" I think is descriptive enough. Or have a section in the program for "Equitation over fences" and below that list the class number and heights.