For PtHA: in an english pleasure class, it is on the rail judging the horse. Hunter hack, is sort of an "S" shaped pattern with 2 jumps.
At the pinto shows, we have show hack. It's almost a mix of an english pleasure class and dressage (even a double bridle is used) but is all done on the rail.
If you go to an APHA show, they don't have english pleasure, they have hunter under saddle. Yet hunter under saddle at the pinto shows has a hand gallop. Go figure.