The key to the depart is to persuade the horse to take whatever rhythm it had been in and change that rhythm to the three beats of the canter. As you know, the first beat of the canter is the outside hind (followed by the diagonal pair and then the inside, leading foreleg). But whether you're in walk or trot when you want to ask, that first beat still comes down the same in the new gait. What has to be altered is the second beat, meaning the inside hind leg which now has to do something different than it had been doing. So when you should ask is when the inner hind is supposed to push off, remembering that your aids include scooping him up with your inner seat as well as what your legs tell him.
Your outside leg is going to be back, but which leg you emphasize as the immediate "cue" will often depend on what they understand. I like to think that on an educated horse, it's my outside positioning leg that tells him which lead to take and it's more my inside leg that tells him when to take it.

FOLLOW UP

Kayla – That’s a great, perfectly sensible explanation.  The few times that I got the transition dead-on, I felt that the outside hind was down, then the inside hind-outside fore came down effortlessly for the transition.   But, although I felt it at the time, I couldn’t visualize what was happening.  You are right about needing to understand both the Betty Crocker method as well as the touchy-feely, intuitive method.  I don’t think you can effectively solve training issues without both. 

One point of clarification:  You know the old hackneyed movie bit where two guys are going to, say, jump off a cliff on the count of three, so they start the count, then one guy interrupts and says, “Wait, so is it one-two-three and we jump ON three?  Or is it one-two-three, THEN we jump?”  That’s how I feel about WHEN. 

From the walk, when the outside foot comes down (last beat of the walk/first beat of the canter), the next beat has to be the inside hind-outside fore (the second beat of the canter.)  So when his barrel swings against my inside, “asking” leg, that’s when I ask?  Or do I have that backwards?

Bill - Well, just as it's ready to swing away from your inside leg. That's when the inner hind will come off the ground, and you want to catch the horse right before that happens.

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