No you're not. When I was learning to canter my trainer told me to imagine I'm polishing the saddle with my butt if that helps lol, I sometimes do 'rising canter' when I'm doing cross country as it helps to steady my horse up but it's not really something you do in dressage or flatwork in general. If they were in two point they were probably either jumping or going for a gallop. You're martingale is to keep your horse from putting their head up not for you to balance on, if you need help either put a strap on (we use old stirrup leathers round their neck to hold onto) or you could just hold onto the front of the saddle. Hope that helped
When you canter, you should lean slightly back, but not too far. If you lean forward and the horse suddenly moved in a different direction than you expected there is a high chance that you could come clean off. But if you say deep in the seat, the would still be a chance if you falling off, but it wouldn't be so high.
IF you hold onto the martingale you will tilt your upper body forward which will position you in front of the horses centre of balance but technically you could you you get nervous. The best place to put your hands is to carry them yourself or to use a monkey grip on the front of your saddle as you can still maintain your position
When you can't in English the goal is to hold your self in the saddle. This takes a lot of core strength. You ARE NOT supposed to bring your self up out of the saddle. The 'up and down" motion you mentioned from videos of other people riding is incorrect. That motion is actually very bad for a horses back. Tighten your abs and sit back. Try completely relaxing your lower half of your body and allow your horses motion to slide you in the saddle just a little bit. But try your best to stay in one place and it the saddle.
Sydney, your pelvis is your butt. having rode both english & western for 15 years, I know the difference in your pelvic position. I can also tell you how the pelvic position alters the horses movement. & riding finished horses that compete in both english & western, I can tell you how the pelvic position determines the speed of their gaits.
Yes sadly this is because they oftentimes haven't got the core strength to keep themselves out of the saddle and therefore cannot hold themselves in a correct 2 point position but the rider I see not internationally connecting themselves with the saddle, the horses movement is just affecting their balance . Have a look at Evention TVs video on how to hold your position when galloping on youtube
Oh and for you hands in a canter in English, you would like them to create a line from the horses mouth to your elbows so they are free to move but your you are not confident put a monkey grip or piece of bailing twine on the front of your saddle and hold that with one hand and the reigns with the other.