Years ago I was taught to ride 'Western' by an old, bent bow legged Canadian cowboy who had been involved with horses since he was a kid. In the 1930s he had been a winning rodeo rider. By the time I met him he had formed a Western riding club in Surrey, where anyone who rode horses used the English hunting seat. Kennie's first job with new members was to teach them how to ride Western on his Western schooled horses.
The first lesson was to adjust the stirrups so that the leg was carried almost straight. Enough bend was left in the knee to just lift the butt off the seat of the saddle even at the trot.
The second lesson was to learn to ride with signficant weight carried on the stirrups at all times.
The third lesson was to move with the horse, if it leant over, then lean with it.
The rider sat upright and straight using the feet to compensate and resist the
Forces of gravity and movement by pressing down on the stirrups - which were almost being used as 'pedals'.
The rider leaned with the horse - if the horse went to the right at speed then the rider would lean over with the horse into the bend.
We always were to ride on a loose rein held in one hand only. The bits were all Western lever bits and we were told never to ride collected as the potential for accidental pressure on the horse's jaw was too great.
When we trotted - which was usually on level tarmac - we always posted.
If the weight was held on the stirrups, then the rider could not rise too high so long as the stirrups had been adjusted correctly for Western. The knee joint took the strain of rising to the trot.
The riding technique as described above was regarded as almost heresy by regular English riders trained by the British Horse Society. But it worked.
My horse would accept being ridden English or Western.
With hindsight it would be interesting to video a western rider posting and an English rider rising to the trot. If the camera were high speed and could be slowed down then the difference in posting and rising could be better understood. A high level English dressage rider might already know the difference.