I'm not a pro anything, but FWIW:
With our mustang pony, if you put the saddle too far forward, he'll buck. With the other two, if the saddle is a little too far forward, they get pissy in trots and turns. They will obey, but you can see they are not happy.
If you get the saddle too far back, then if you bounce on a sitting trot, they will get pissy, although they behave better when posting or walking. By pissy, I mean they will pin their ears at the start and be eager to slow to a walk. My larger horses are not buckers.
I have a couple of English saddles that I rarely use anymore. The forward flap of the jump saddle should slide over the shoulder, not dig into it.
Bouncing at the trot: I doubt it is the saddle, but it could well be your stirrup length. I don't jump, other than the sideways jumps my mare sometimes throws in to see if I'm awake. I ride with a long leg. The longer my leg, the easier it is for me to sit the trot. However, I got my stirrup straps so long that I started pointing my toes down when cantering, so I just moved them back up a hole last week. As my stirrups get shorter, I find it easier to brace against them, which makes me bounce more. It is possible to ride with short stirrups and a relaxed leg, but it would take me a long time to get used to it.
Also, the easiest way for me to sit the trot is to have my feet forward, not under my hip. That may be how you ride bareback...lots do. Something about pulling my heels back under my hip makes me bounce more. Maybe it is the tension caused when I pull my heels back. In any case, since I don't do any horse sports, I normally ride with my heels below or just in front of my belt buckle.
So I doubt it is the saddle by itself, or its location on the back. My guess as a beginning rider myself
is that it involves your leg position and tension in your legs muscles.
The picture below is my normal leg position. As you can see, it would be way too long and completely wrong for jumping. I'm a very *******ized rider, so hopefully someone who rides English and knows what they are talking about will chime in.