The second manner describes by Baucher is very interestning. He developped it after his accident in 1855. As he had quite lost powerfull in his legs.
Some poeple in french Haute Ecole still use this way, including what they call flexion of the jaw.
This flexion has a really strange result if you listen to a rider using it, you'll hear the bit (these people never use briddles) making a sound in the horse's mouth which has something to see with the shock produced by a jerk but it is not produced by a jerk since reins are not tightened.
Infact this sound is produced by the bit moving almost freely between teeth, tongue and palate.
That's why it's conception is made of many pieces to allow movement in the mouth(on the opposite with a briddle which is a one piece metal in the mouth).
It was really strange for me to realize that the horse was collected without tension of reins, still hearing the sound of the bit without any drop of saliva. It was completely on the opposite with what you can learn in dressage according FEI rules nowadays.
I have to notice that I just know very few people able to achieve that with a bit and I'm sure not to be good enough to test it on my horses.
I prefer going on without bit.