I would caution you against what Hidalgo is suggesting, you may turn one bad habit into another and create a horse that runs out. Another good alternative is to trot to the fence and immediately woah afterwards, or take the jump on a circle so she has to think about where she's going afterwards, and doesn't have time to get into 'rushing' mode. Grid work can also work wonders.
Although watching your video, your canter is way too big and fast WELL before the jump, I think this is a problem that needs to be solved on the flat first. If you have a generally rushy canter, you're not going to be able to slow her down approaching a jump. I suggest working on control on the flat FIRST.
do agree that many of this horse's vice could be fixed with flat work. As for your position, it is such a short video it is hard to judge. And, as said, bareback will not be the same problems as riding in a saddle.