Well, the term can indeed be applied to mean different degrees of what Ed Cornell explains in his books on Hackamore and also his book on Vaquero training.
This happens with many terms, Bluey Another example is the term ‘spur stop’
Many people use it just to describe a horse that is spur trained, and also able to be ridden mainly off of leg and seat, but the true meaning of that term, ‘spur stop’, is a controversial method that was /is used by some western pl trainers where that spur is held constantly into the horse, to rate his speed, to the point that spur marks are left behind and the horse thus becomes a ‘one trick pony’, intimidated to move out
I debated about using this example, as it will further fuel negative impressions against ALL western pl horses and trainers, but at the same time, I am aware of abuses not just in other disciplines, but also within those that I show in
The original true meaning of doubling, was as described by Ed Cornell, at a time little if any ground work was done, and thus used to show a GREEN horse that he could be controlled with a bosal
Of course, eventually that respect of the bosal, along with training, produced a very light and handy horse.
Even today, methods are used that can be considered harsh, and forceful, to get a concept across, versus gradual teaching .I have seen reiners fence a horse, where the horse actually hits that fence at full gallop. It does teach them quickly to stop on a loose rein!
Perhaps the doubling in Spain, that Ed refers to, was only done at a very distinct area, and I don’t think he was implying that it was used in the general training of Haute Ecole
Certainly I never heard the term doubling used in western training, but am familar of course , with the term rollback, or, used for control -the one rein stop