I don't know about what designates one level from another, but the main difference between hunters and showjumpers is that hunters are judged primarily on the horse and his performance and way of going. Jumpers are judged on clear jumps with minimum faults and fast times. Equal runs are decided by jump offs where the fastest, cleanest course wins. I generally understand the difference at its most basic to be hunters kinda like pleasure, and jumpers more like barrels in the way the classes are judged. Hunters are a lot more "traditional" in rider turnout, etc., as far as the actual appearance in the ring. It's less unusual to see brighter colors (for English, lol) in the jumper ring, maybe even colored yarn in braids or even unbraided horses, at least in the U.S. And in my experience.
Your mare is so fortunate to now be in a loving home with the care she deserves.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown