it's great that you are looking at becoming a better rider in every small detail.
learning to feel the belly roll to the side is good because this indicates when the leg on that side is stepping under. So, if the belly swings away from your left leg, that means the horse's left leg is reaching under him. This means that if you want to cue the horse for either a sideways step (away from the left leg) or ask for more forward energy, then you time your application of your left leg WITH the horse's stepping under/forward with HIS left leg.
I do not apply my leg with the roll of the barrel every time. this would dull him out.
if you want a general concept of riding that is real fundamental to riding I would say, Don't put any aid on unless you want a change in the horse, and don't take that aid off until you've got a change in the horse.
a lot of problems occur when folks are putting a leg or rein on kind of mindlessly, not looking for a change and certainly not noticing when it happended and rewarding with a release.
The opposiste is ; asking for the horse to do something (applying an aid) and then releaseing the aid when the horse has not done what you asked.
both of these train in dullness.
Take stopping for instance. I ask for a stop, that means the horse stops and comes off the bit and even rocks back a tiny bit on his haunches. if I release when he is stopped but is still leaning forward so much, barging on the rein, that if I release the reins he falls forward and off he goes, then I am not getting a real change in the horse.. I release only when he has fully stopped and has changed his mind and released forward.
if , when he is stopped, I keep the rein tight and never release, I am applying an aid for no reason at all, and making the rein have less and less meaning.
same goes for the leg. If you put the leg on and the hrose doesnt' make a real change, and you release the aid, then you are saying, "that's ok, I 'll accept this half response" from now on, that's what you'll get.
but, if the hrose is already trotting along and you keep naggin at him, he'll learn to ignore your leg.
So, that is a really important principle in riding, from my perspective.
I read the Sally Swift book years ago and it gave me a lot to think about. It's a classic.