I do not have time to read everything so someone may have already explained it this way:
One utilized the diagonal aids
and the other utilizes the lateral aids.
Diagonal aids means that you are using a dominant left leg and right dominant rein. This asks the horse to move his shoulders and 'follow his nose.
Lateral aids mean that you are using the dominant lag and rein on the same side. In the beginning, the horse;s head will be turned slightly toward the dominant leg and rein. In doing so, you steady the horse's position and ask him to move his entire body away from the dominant leg aid. Your opposite rein helps to steady your horse and keep him in the correct position.
Your aids for side passing are the same as for teaching a leg yielding exercise while you are moving forward. I prefer to teach a good solid leg yielding response before I teach a side-pass so that horses get in the habit of crossing their feet over ahead of each other instead of behind each other. Then, they learn to cross over 'clean', coordinated and do not interfere or hit the other foot or leg.
The object should be to use less and less leg pressure and to gradually get the horse to keep his head straight or almost straight in front of him.
The position of the leg is the same for leg yielding, side passing and turning. The leg should move back slightly to ask for a turn on the forehand or disengagement.