I'm gonna steal her thunder and answer... Cutback saddle... named for the cutback at the wither.
Also called a flat saddle or "saddle seat" saddle, it's the type of saddle used to show horses that are not 'hunters'... IE, Arabs, National Show Horses, Gaited Breeds, etc. It's just a style, styled after the Southern country gentleman's plantation saddle, rather than a jumper saddle. People show in formal long coat-tails and derby hats, rather than the shorter hunter jacket/breeches. And the horse is flashier, showier, with a higher tighter headset and higher, snappier action.
Years ago in Arabs the English Pleasure horse was a saddle seat horse, with the kind of pretty action and head carriage we all picture with Arabians. The fad when my stallion was about ready to retire anyway was looking for MORE and MORE action, to where "English Pleasure" became "BIG NOISY TORNADO OF A PARK HORSE". There had always been a Park Horse class, but those horses started to dominate in what has always been the quieter Pleasure class. The older traditional English horse, the nice pretty, showy but quieter horse that actually looked like a pleasure to ride through the countryside on, became "Country English Pleasure". That's the type cjshapi is training her horse for. It's also the type my stallion would be in today.
Personally, while I marvel at the movement and fire of the Park Horses, they aren't natural, and I don't think they are the best example of an Arab I'd like to own. Not now, I'm quieter now. :~D