So far the critiques have been right on. And I agree, the horse is cute.
You are struggling with how to handle all the movement the horse provides. We see good riders riding so very still on their horses and we think, "oh, it's so easy. All I have to do is sit still". Aux Contraire! YOu have to move. YOu have to move to sit still. And you may have to be using some muscle to move, too, not just be flopped around.
If you were to strap a manikin on the back of the horse, as the horse moved, the mannikin would be sloshed back and forth, and the more the horse moved, the more the mannakin would whip back and forth. In fact, it really is "whiplike", the way the motion of the horse moves through any thing that is on there body. The saddle would whip around, too, if it weren't so tightly held to the body, and it's center of gravity so very close the horse's cg. Since you are not "held" to the horse, you must create a kind of suction or connection of your seat to the saddle/horse. I dont' mean grip like a leach, because what happens if you do that is the the rest of the body becomes so loose that IT flops around.
Rather, by making your lower body move exactly WITH the saddle, you are creating adhesion. IN order for your pelvis to follow the saddle, it must be very loose and supple in the joints of the lower back and the hip . You have to really release your lower body to let it go with the horse.
At first, since you don't know how to "let" it follow, you must "make" it follow. Exaggerate the movement of you lower body for a bit and think of your seat bones as 6 inches INSIDE the body of the horse.
Once you can move with the horse with your lower body, you pay attention to the upper body. YOu need to sit up really straight, engage you core muscles and keep your elbows near the ribcage to disallow the upper body to flop. You basically "ride " your own pelvis, which is "riding" the saddle/horse.
The motion of the hrose is absorbed in your body, thus your hands can stay in the same place relative to the horse. You must have movement in the elbow in order to be able to do this. One thing that helps is to think of bringing your core toward your hands instead of the other way around, as the horse rythmicaly moves.
All this can best be worked on while NOT holding reins, so basically, a lunge line lesson. If I were your coach, I would have you do more than a few lunge line lessons, with no reins. Just focussing on moveing with your horse. Once you get this, you pick up the reins. ONce your seat is stronger, you will be able to hold the reins quieter and your horse, who seems like a very kind and tolerant fellow, will be so much happier to not have you bopping on his back and mouth. I bet he'll not be rushing out in the trot and will just be a lot nicer to ride.
Your place in the journey of learning to ride is not so different from many, many people who learn to ride. Some NEVER learn to move with the horse, so kudos for you for wanting more and better.