Some people feel it is easier to learn the rising trot before learning the sitting trot. I think this depends, in part, on how the horse is moving. It is easiest to ride rough or energetic trot by posting.
Unless you can get a livelier trot out of this horse, I would suggest trying the sitting trot first in order to better work on your balance. Then, you would have a better chance of learning to do a good rising trot. You would also have a better chance of getting a more active trot from this horse.
It appears that you are sitting too far back in the saddle. Try sitting in the middle of the saddle with your head balanced over your spine. There should be a basic vertical alignment of head over shoulders over hips over heels. Think of standing with your legs apart and knees slightly bent. A horse just happens to be between them.
With your pelvic bone vertical so you are sitting on your crotch as well as the seat bones, think of balancing your head over your spine rather than sitting erectly. If you relax your muscles, the bones of your spine should stack one above the other forming natural shock-absorbing curves. Your bones, rather than your muscles, will support your weight.
Next, release tension in the muscles around your pelvis, throughout your crotch area, all the way down your legs, and even throughout your ankles, feet, and toes. It may help to do this first with your feet out of the stirrups.
As you release your muscles, you should feel your seat sink deeply into the saddle. Gravity alone should rap your legs lightly around your horse without any squeezing on your part. It should feel as though your legs would drop to the ground if not attached.
If your feet are out of the stirrups, your toes will hang lower than your heels. If the balls of your feet are on the stirrups, the stirrups should support your feet without you pushing down. Gravity should pull your unsupported heels lower; don't worry about pushing them down.
Practice walking like this first and feel your body moving with your horse. Then, ask for the trot with a light squeeze and release of your legs. Don't forget the release or your muscles will be tight, and you want them relaxed.
If you are well balanced with a low center of gravity, you should not be thrown off balance when your horse begins to trot. If your arms and shoulders are relaxed, your hands should follow your horse's mouth, and you should not pull on the reins. If your muscles are relaxed, your seat should ascend and descent with your horse's back.
If you tighten the muscles in your crotch, your center of gravity will rise and your balance will become unstable. If you squeeze with your legs, you will block the flow of gravity as well as inhibit the movement of your body which should be following the movement of your horse. If you hold your breath, your muscles will tighten.
Remember, balance and relaxation are keys to good riding.