Gotta chime in. Your seat is correct or not regardless of the saddle. I don't know many people who do NOT start their horse on a snaffle bit today, although Western riders usually switch to some kind of a curb at some point, and English riders often just keep their horse on a snaffle throughout their career.
The Western saddle and tack have evolved from Spanish explorers and Mexican Vaqueros. You can see it with the tooling. The flowers and geometric shapes on Western tack are based on Islamic decoration, from the time when Spain was part of the Ottoman Empire. The pommel and cantle are higher bc the original Spanish Explorers in North America and their descendants kept this in their saddles. In fact, the swell on the Western saddle is a 20th century invention. The late 19th century Western saddle had a more tapered pommel, still deep, and rightly called a "Bucket Saddle." Alt View
The horn evolved as a way to tie off cattle, hang ropes and things and generally aid in cattle ranching, which brought beef to eastern markets, which of course brought back the western American economy post Civil War. (1861-1865).
The English saddle was a simpler, lighter saddle, brought over to Colonial America (1607-1776) from, well, England and Northern Europe. It didn't require a pad, and it's derivations enabled fast travel between Colonial and post-Colonial towns, as well as jumping downed trees and the like. In the late 19th century and early to mid-20th century America, fox-hunting was a popular necessity and evolved into a popular hobby. The dress and tack for fox hunting has become the dress for Hunter/Jumpers in the show ring. A curb bit isn't necessary for jumping. Plus, you don't want to catch your horse in the mouth accidently and also pull his head down (with a curb chain can do) as you're riding over a jump. However, it's much easier to quickly change directions while cantering or galloping if you direct rein your horse.
MOST everybody in America eventually rides their horses both English and Western, depending on where they are riding. I've know a LOT of horseman who show English, but pack Western tack for a trail riding vacation--SAME horse. Full day rides do not lend themselves to an English saddle, IMHO. Plus, Western saddles have places for big saddlebags, ties for a slicker, places to clip/tie/hang extra canteens, etc.
I won't go into what are the best breeds for what, but I think we'll agree that Western-ridden horses should have a comfortable trot that you sit. (I knew an old cowboy in the 1980's, and he told me that "A cowboy wants a horse with a good walk and a good lope. Cowboys don't trot their horses."
) English-ridden horses should be able to jump small to medium hurdles, and you post the working trot. (Saddle horns get in the way when you post.)
And, gaited horses...um...well...they DO make cut-back Plantation saddles...I own one, so I can ride the big, gaited horse to the left, in my Avatar. I direct rein AND neck rein him, btw.