I'm starting this post with a confession. Saddle seat is something I don't know as much about as the others.
Saddle seat is how you ride a gaited horse. Gaited horses move a bit differently from non-gaited horses. You can't ride a gaited horse in the forward seat (even the slightly less forward version used by dressage riders). I learned that in Iceland, with apologies to my horse. Fortunately for her, I got it quickly.
The saddle seat saddle is positioned slightly further back to allow extra freedom of movement for the horse's shoulders, needed to gait correctly. When it comes to the hands and the bit, however, saddle seat is just like English - the horse is ridden on contact, but the hands are held higher due to the higher head carriage of a gaited horse.
Saddle seat is worth studying (and trying) if you're writing medieval fantasy. As a means of transport, gaited horses are superior to non-gaited horses and saddle seat is the closest to how people rode back then. (Icelandic saddle seat, in fact, pretty much is how they rode back then, and the horses are pretty much what they rode back then. If you can track down a local Icelandic breeder, you'll be golden).
Saddle seat show riders. Note how far back the saddle is on the nearest horse. These horses are American Saddlebreds. Image source: John Goetzinger via Wikimedia Commons.
An Icelandic Horse, Saevar fra Stangarholti, gaiting at speed. Again, see that the saddle is further back, but a slightly different bridle. Those Icelandic saddles are incredibly comfortable. Image source: Dagur Brynjolfsson via Wikimedia Commons.