I don't *totally* understand what you're asking but I'll have a go at answering anyway.
Obviously posting/rising trot, the rider is out of the saddle for one beat of the trot, and in the saddle for the other. In sitting trot, the rider remains in contact with the saddle for both beats of the trot and the moment of suspension.
Rising trot is mainly employed in young and green horses, and in warming up prior to commencing more difficult work on a more educated horse. Riding trot, when the rider is comfortable and capable to execute it - not bouncing on the horse's back, thumping down hard, being left behind the movement, moving hands with the upper body etc., will encourage the horse to loosen and lift it's back, as it does not have to carry the riders full weight directly on it's spine. Particularly when many riders find the sitting trot difficult, and tend to brace against the motion - blocking the horse's back.
In a young or green horse, the back is not yet strong enough to carry the weight of the rider in sitting trot for an extended period of time, and often will hollow their back away from the weight of the rider. So rising trot allows to rider to be lighter and less interfering over the back.
Sitting trot, we do in dressage because this is where we can gain a real connection with the horse. In sitting trot we can use our seat aids to their full capacity, we can feel every movement the horse makes, can feel tension in the back and can follow the horse's movement, giving us enhanced ability to control each footfall.
Dressage tests reflect the introduction of sitting trot, from rising trot on young horses. At training/preliminary level, the rider is allowed to rise all of the trot work. As you progress through the levels (I am not sure on the US tests but here it is Novice level) the rider must sit trot the working trot, and is able to rise the lengthens. Only at Elementary, must the rider begin to sit all trot work. At this point in the horse's training, it should have build sufficient muscle and strength over the back to carry the rider, and should also have the capability to lengthen the stride and frame while lifting the back with the rider in sitting trot.
Also when we reach this level, we begin to introduce some lateral movements. Riding laterals in rising trot is quite more difficult than in sitting trot, and the rider has less control over the horse's 'core' and footfalls. Hence, we sit trot and encourage the horse to step it's inside hind leg, under its, and our, centre of gravity.
I hope that's given you a bit of help with your question!! It may not make total sense - I'm about to head off to ride so typed it quickly... will go over my response properly when I get home.