Point of history: The US Cavalry manual written in the 40s was very much a manual on jumping and the forward seat. It was not, by any measure, a manual on dressage. Nor were any of the previous manuals, because dressage is ill suited to the functioning of a military. A military needs to train large numbers in short times.
Yes, the US Army had folks who rode dressage. It also sponsored a polo team. But the style it taught is pretty obvious in the pictures of the military:
WW1, with dual gas masks:
Not sure of the age of this one:
I believe this is Canadian from around WW1:
The Italians, of course were nuts:
Believe this guy was British:
OK, end of history lesson. But there isn't much evidence that the military was teaching folks a dressage position...