those are such cute horses! :D i love the colors of brass and jake! and southafrica and mirta, huntseat is the style that jeaninal is riding in the third pic. it was created for fox hunting and is very popular now. (thats the style i ride) and saddleseat is more fancy. you wear more showy clothes and the horses hold their heads up high and lift their feet as high as they can. it isnt as relaxed as western or huntseat, you want your horse to be pumped so if performs to its best. i'll give some pics below.
this is a western horse and rider. it is very relaxed and laid back. the speed is very slow and the horses head is lower and it doesnt lift its feet too high.
this is a western saddle. it always has a horn on the front and usually has some kind of silver on it. western bridles usually have more silver on them too.
this is huntseat. it is faster than western and the horse carries its head a bit higher. the rider sits leaning foward just a bit because this is how you would sit if you were fox hunting and jumping over brush.
these are huntseat saddles. pretty plain right? same with the bridals. unlike western the rider wears a helmet when riding huntseat. you also wear a short coat and "breeches" or the tight fitting pants. you also wear boots.
this is saddleseat. im assuming that just by seeing the pics you can now tell the difference between this and huntseat. in saddleseat the horse in excited and ready to go(they have to be to lift their feet that high) they hold their head up and lift their feet. they only go a bit faster than huntseat but only because they are so pumped.
this is a saddleseat saddle. it is even plainer than the huntseat. the only obvious difference is that the huntseat has a pad where you knee would go. you can tell by the pics that the outfit for saddleseat is very formal. you wear a long coat with slacks and a hat called a derbie. you have to hold your hands higher than you would have to in huntseat or western.
hope that helped