In his book, 'The Ethics and Passions of Dressage' (yes, again!) Charles De Kunffy writes about the "three different most basic hand-rein positions, on which all others seem to be refinements or elaborations."

THE DIRECT REIN
He says that the direct rein "has the most thorough and direct influence on the haunches of the horse. He can feel in his mouth exactly what the rider is doing with his seat. The rein's direction remains as straight as possible between the the rider's hips, through his elbows to the horse's mouth." He says that "when such a rider's seat follows the motion, his hands automatically contribute to engagement by 'opening the doors of progression'. He says that "As the horse is advanced by the schooling rider in his mental and physical development, the rider earns the privilage of using a direct rein on both sides."

THE INDIRECT REIN.

He says that this rein "primarily increases the horse's positioning and bending of his neck, but also facilitates bending in the rest of his body." The indirect rein "can only be used on one side at a time ...because the direction of the indirect rein is from the horse's mouth towards the rider's chest or shoulder on the opposite side. Thus the power of the indirect rein is diagonally effective and is only unilaterally useful." and "The indeirect rein is very close to the horse's neck and even touches it."

THE LEADING REIN

This is the really interesting bit as regards AK's system. He says that this rein "is used in conjunction with the direct rein on the other side. The direct rein forms a triangle with the lead rein by distancing itself away from the horse's neck. It is the rider's lower arm, not the upper arm or elbow, that creates the distancing of the rein." He says that "The leading rein is most often necessary on a young, green horse with balance problems and who therefore, drifts. Drifting of the shoulders can best be counteracted by the use of the leading reins. The rider lowers his fist position...and forms a triangle with the reins...

This sounds just like the single rein riding in 'Riding With The Clicker' to me.