A gentleman would have hack horses, for riding around from place to place, a hunter, specifically used for the fox hunting (bred specifically for this job and used exclusively for it--you would ride a hack to the meet to spare the horse for its job of hunting), and, if he wished, racehorses (Thoroughbreds, bred by Charles II for that purpose).

Since foxhunting was a very new sport, I don't think that the breeding of hunters specifically was much developed in JA's time. The hunters near her home in Hampshire seem to have used any old horse that they could ge their hands on for it. Jane's brother s went out on ponies and Squire Chute, their hunting-mad member of parliament just grabbed a horse that he hadn't already used up in the week and spent the day on it. ( The older sport of deer hunting didn't involve much riding, and was more like herding and culling than a race across fields). Often a "hack horse" meant a rented horse. But since we have had so much controversy on the meaning of "hack chaise", I'm not going to push that one too far.....;-)

If I understand the sport right (particularly steeplechasing), the gentleman could choose whether to ride the race himself or employ a jockey to do it.

Last march we had a good conversation about the development of the Jockey Club and the lives of professional jockeys. Of course, there were always young men like John Thorpe and Tom Bertram who were willing to race against each other for the fun of it, but the big races were run by professionals.

Fat men would ride cobs, short fat horses who could bear their weight,

Actually, the Welsh Cob, as a breed, was developed as a carriage horse.Though I am sure you are right about Fat men finding sturdy horses for themselves;-)

and by the nineteenth century, many breeds of farm horse existed;

Perhaps, but no-one sems to refer to them by breed. Usually it's just a "horse" or a "plough horse" or something like that. Do you have any quotes to share on this? I'd be very interested to know of specifc breeds being mentioned..I've got a bit on cows , quite a lot on pigs, and something on sheep, but on horses, I have nothing. And that's strange, because even in the breeding mania of the 1750's, no-one seems to have specialised in horses.