This is becoming a pet peeve of mine. Equitation shouldn't be discussed as being independent of tack.
You can ride a jump saddle like a dressage rider, but you will be fighting your gear. You can ride a dressage saddle over jumps, but you'll be fighting your gear. And the switch to a western saddle is even greater.
Ride on your seat pockets in an English saddle, and you'll hurt your horse, who will respond by hollowing his back and getting upset. Do the same in a western saddle - which distributes weight to the rear over a much larger area - and the horse will bring his hind legs under a bit in a modest form of collection.
A chair seat makes sense for cutting cattle. It isn't right for dressage. A leveraged bit can work fine with slack reins, but shouldn't be ridden with constant contact. You cannot separate the goals of a particular style from the equipment used and the style of riding itself.
Rather than running everything thru the blender, we ought to appreciate and respect the differences. The goal should be stew, not pureed mush. I don't have any desire to ride Dressage, but can appreciate and respect those who do. Western riding has its own history and traditions, and the tack is designed with that in mind.
As has been mentioned, reining takes some of the principles and approaches of dressage and reinterprets them in light of western gear and history. It is true to the spirit of both. "Western Dressage" is, IMHO, an insult to both traditions while appreciating neither.
Frankly, I think the folks pushing western dressage don't understand western riding very well. The explanation below reeks of condescension to western riders. It is as if we are all a bunch of idiots who can't even get our horses to stop and relax or to travel straight or turn well - all of which is bunk!