I think there is another version called a french braid that keeps it on the crest but I have never actually seen it done. Exhibitor Labs : French Braiding The Long Mane
As for hunter/jumpers, no, you would want standard hunter braids. A running braid would get in the way of the crest or open release needed. Running braids tend to get used more in the dressage arena.
I tried this recently on my TB I admit :). He had a nice long mane and I really didn't want to pull it to put up in dresage braids..I tried a running braid but his mane was just too thick.
What I didn't realize is that there is a new style of dressage braids which I have yet to see while being created. I have seen the results though :)