How I've been taught and how I taught my horses -
For dressage, the leg, seat and rein aids all work together, as you noted.
walk, trot, canter, leads, turns, rein back.
In dressage, we ride a lot of half halts. Half halts are a use of the leg, seat and rein, nearly at the same time. They are kind of like conversations, they can be very 'loud' if a horse is running away or very 'quiet' if the rider just needs to let the horse know something is coming up. If a horse is very well trained you almost just 'think' half halt. If it is a bit of a bully or getting really naughty, half halt might be rather forceful. But the idea is to be as light as possible, and always give them a chance to listen to a lighter aid.
The seat aids are not tucking the pelvis, 'sitting on the pockets' or rounding the lower back or 'pushing' or 'lifting' with the seat. No 'humping the saddle', LOL.
There is a type of half halt to just alert the horse that something is going to happen, and would be using half halt on any type of transition(change of gait or within a gait), upward or downward, and probably more of them on any downwards transition.
I'm not going to list every single half halt I would be doing, as it would get too repetitive. But they are 'automatic', 'all the time'.
The dressage aids really are always kind of the same for everything. It would take a lot more words to explain why. But they are not as complicated as they sound. And the aids start out as more obvious and then just keep getting more subtle (harder for an observer to see) all the time.
Walk. Depends. From a halt? or from canter? trot? From a halt, a light squeeze of both the calves right at the girth, a very slight softening of the contact, without dropping the rein contact.
From other gaits, stop following the motions of the gait, passively resist the motion...'sit tight', 'holding' the seat, just sit still.
If I want to walk from canter, sit very, very still and firm, refusing to be moved by the horse's motions, 'hug' the saddle with the thighs, kind of like 'holding your breath', half halt with leg, seat, rein.
Trot. From walk, urge with both legs at the girth. Some people use a couple light taps with both legs to ask for trot. I never found that necessary.
Canter. On a trained horse, push the inside seat bone forward and very slightly 'open' the leg at the knee, keep the contact with the reins, almost just 'think it'.
On a less trained horse, outside leg slightly back an inch or so, inside leg used at girth. On a little more trained horse, half halt outside leg is hardly needed, inside leg at girth, a very slight 'softening' of connection with the inside hand.
To rein back, I sit very lightly in the saddle, even with the upper body very slightly inclined forward, so a small amount of weight is taken off the back of the saddle and the horse can step back easier. Both legs back behind the girth, and gently use each rein.
To get the leads at the canter, from the very start I try very, very hard never to teach the horse to pick up the lead based on which way its head is turned. Most people, if they want to pick up the left lead, they bend the horse left. I don't do that.
I try to keep the head and neck very straight and teach the horse to take the lead based on the leg aids, just a nudge with the outside heel to start with. I don't use a shift of weight, either. I don't want to affect the horse's balance later on so in harder movements, so I don't do that. There is a very slight adjustment of the seat that follows the lead change, but I don't want to shift any weight to get them to do a flying lead change.
That is because later when I am trying to do series of flying lead changes to a count, or zig zags, I really need to not be wagging the horse's head back and forth or shifting my weight, and affecting his balance badly.
To turn. As if it is a part of a circle, same aids as for a circle. Circle aids - Outside leg behind the girth (just as a 'guard' to prevent haunches from swinging to the outside). Inside leg at the girth with inside rein to help the horse bend. Outside rein to keep the neck from bending in too much or the outside shoulder drifting off the line I want to be on.