I don't pick up either rein, I maintain a feel with the outside rein, keep the bend with my inside leg, soften inside rein, and cue with outside leg back. I don't think it's a good idea to train by lifting a hand, just keep them quiet, and keep the horses jaw soft, but ride from seat and leg. If the horse stays soft in the jaw, then they don't brace their jaw and go stiff in their necks and backs and can use their back end to engage and drive.
I would think that the transitions go through a process for horses and rider's though, and what you start out feeling is going to change as you get more balanced in your seat, and more connected, with a better feel of how your horse moves. On more advanced horses, they might pick up the right lead with just a shift of the seat. My 1st level horse will pick up canter from walk on a loose rein with just a little outside leg back and a light bump with my inside leg. He schools counter-canters very nicely and can pick up a counter canter from walk on a 10 metre circle. My green tb is at the stage where he needs a lot of help and preparation to get a nice transitition into canter, and his balance in trot is very irregular so I spend a lot of time on the 20 metre circle working on rythm, and engagement, and response to a half-halt before I can apply the canter aid and have him pick it up nicely. It's coming, and when we really work through a schooling session, by the end of our session, he is picking it up from walk. I do believe though, that he never was very good at picking up the canter under saddle until he was really getting it nicely on the lunge line. In our last few sessions, his contact with my hand has gotten very steady, he used to curl up and try and avoid the contact, and was tense under saddle. Lately, the light bulb has gone on for him, and he is so relaxed and calm under saddle, and his attention on me is solid, where he used to distract on everything around him. I was soooooooo excited after our ride today, because I finished up by cantering over a cross-rail and he landed on the lead I asked for each time, and he's never done that before, and I haven't jumped him at all in the last month, so that is all about his balance and connection and responsiveness that he's gained from his flat work!!
Whoops, not wanting to steal from you and your thread.
Would like to hear more about you and your horse.
Robin