Well, the pictures get better as they go along. There's a bunch of things you might try to help build this into your muscle memory. Because you "have" it at the walk but loose it when trotting or more, right? So you just need to work on building that muscle memory. And really, all of riding is doing just that. If you learned how to post, how to keep your foot in the stirrup , how to keep you chin up, how to hold the rein, etc, you can build this , too.
One thing I see is that when you are in the upmost part of your post, your leg is nearly straight. I would consider shorteingin your stirrup just one notch. Your shape and your horse's shape make for a tought one for you to get your legs "around" the horse, but having post straight legs at the trot means you no longer have as good a base of support as would be helpful.
You see, you have to be secure and stable in the body in order for you to be independent in your hands and able to put them where they should be and let them stay there, while the rest of you moves around. I think the one reason that you feel more capable of doing the bent elbow after gym exerxising (which leads to help keeping the thumbs on top and a "quieter, more following hand overall") is not that your belly is smaller but rather that your core is stronger.
Having a good strong core is just absolutely fundamental to good riding. Not just strong abs, but strong back muscles and the ability to keep them engaged. If you ride really thinking about moving your core "toward" your hands as you post up and down, you will be better able to keep them quiet. Almost think of moveing toward and "through" the space between your steady hands.
Keeping a bend in the elbows is also very much assited by keeping your chest open and thinking of having your elbows stretch down toward your pelvis. Try to bring your elbow closer to the top of your pelvis, and keep those elbows "riding " your own pelvis. When you do this, you will feel a "tug" at the collarbone area, as you open chest.
So, open chest, engaged core and back muscles, post TO and THRU your hands.
One more thing;
When you keep your thunbs on top, think of the tip of your thumb as laser sighting toward the bit rings. IF your horse raises her head, you must also raise your hands NOT drop them. But that's a whole 'nother post . . .