Sorry this is late!

Knights and the later regiments of horse rode very differently.

Knights rode in a very deep seat, in saddle with a high pommel and cantle. They had long stirrups, and you keep their legs forward. Typically, they would sit the trot. For pleasure, they would ride horses that were somewhat similar to gaited horses. But, they would always sit the trot.

Cavalry regiments of the Napoleonic era started experimenting more with shorter stirrup lengths and the lighter, more forward seat meant for riding across country (or, as we say, cross country). Riding long (i.e. long stirrups) was seen as more proper, but when training a large regiment of cavalry they would typically teach the soldiers to ride short (with shorter stirrups).

When riding short, soldiers would post the trot because it made riding longer distances somewhat easier. However, much cavalry movement was done at the walk. Sometimes, even a charge would be done at the trot.

Now, (DISCLAIMER) this is somewhat apocryphal, but I've heard it repeated a few times by word of mouth, and so I'll add it here for flavor. When militaries would ride long distances, the soldiers would change their rising diagonal every 15 minutes to allow for even wear on the horse's hind legs.

That said, it would not be implausible to see officers and those with extensive riding experience sitting the trot. But, many cavalry men were tremendously bad riders, and the horse was basically a way to get them to the fight quickly.