Fox hunting -- as in hunting an actual fox -- is banned in the UK now.
I don't know the etymology of the terms "English" and "Western" and know less about the development of the tack itself, but nowadays it is a social convention to describe the kind of tack you put on your horse and it gives some indication of what type of riding you do. If you say you ride English, the community of horsepeople knows what you mean because these conventions of speech, even if they don't make all that much etymological sense now, have a social meaning. If another horseperson asks you what you ride, you probably give more specific information, such as dressage or reining or jumping or whatever. But if a non-horseperson asks, the answer is often "English" or "Western" unless you really want to explain dressage to the layperson in question.
That said, since moving to the UK I have had questions like, "You ride Western right? All Americans ride Western."
To my knowledge, "hunt seat" is the same thing as "jumping position," "two-point." It is the seat you would have if you were galloping across fields and jumping things. Did not the show ring disciplines of hunt seat equitation and hunter classes start from actual fox hunting, when the original point of the show was to judge the suitability of the horse and rider for fox hunting? Presumably they have hunter classes on this side of the Atlantic as well, but I'm so out of the horse showing loop here I haven't clue how those compare with the ones in the US.