What follows is from a backyard rider - ie, someone who doesn't teach, compete, or show. My horses don't do flying lead changes unless they decide to do them, only sidepass when it makes sense to them, etc. So take this with a big steaming cup of FWIW:
Ultimately, to canter you need to convey the idea of 'go faster'. A horse trained properly - unlike mine - can also be trained to take a left or right lead when doing so. But when you are doing something for the first time, doing everything 'properly' can interfere with doing it at all. At MY level of riding, just getting the basics done can fill up my available brain cells.
If your stirrups are too long, you'll feel like you are reaching for them. But if you grip with the knee, and don't allow weight to flow without interruption to the heel, then the stirrups will still come loose. So if you don't feel like you are reaching for the stirrups at a walk, you are probably gripping with the knee (or thigh).
But you can also have problems if you are busy doing things with your lower leg. If you need to give animated, big 'cues' to get the horse to obey, then A) it will ruin your riding, and B) it teaches the horse to ignore the rider's cues - and I think that happens often with lesson horses.
At MY level of riding, I wouldn't worry a lot about cueing the right lead - not on a horse who is resisting going into a canter anyways. I prefer trails to arenas, so the correct lead doesn't mean much to me. We're probably going relatively straight, and any turns may be followed by a turn in the opposite direction, but at a canter none of the turns I do will be very sharp anyways.
Rather than worry about applying pressure at just the right point with one leg at X and the other leg at Y while tipping the nose Z degrees, try just giving a cue to 'go faster'. Squeeze with both legs, verbal urge - and if the horse doesn't obey, then hit your own leg with a whip or crop. If need be, then use the whip or crop on the horse. But rather than worry about telling the horse which lead to take, emphasize "Go faster - now!"
The advantage to the rider is that it is easier to stay balanced and engaged - to stay WITH the horse, and deep in the saddle. The "neutral position" for my legs is heels at the belt buckle. If asking the horse to go faster, I'll slide my heels back under my hip. (Once we get going faster, they go back to the neutral position.) But I'm not moving my leg away from the horse to kick it, and instead am squeezing both legs together in the same location. That makes it easy for me to stay balanced and comfortable and thus relaxed
- because tension in my leg and bracing with my leg are the two great evils I battle when I ride.
If I try to move my legs to different spots on the horse, and particularly if I then need to move my leg away from the horse to provide some kicking power, I find I just set myself up for failure.
The big drawback I see for you would be in doing group lessons in an arena, since cantering on the wrong lead is quite bouncy, and a group lesson doesn't allow you to change directions at will. OTOH, a lesson horse knows darn well what lead to take when cantering in a circle, since they do it all the time (unlike my own).
Feel free to ignore most of what I've written. As I said, I don't teach or compete. But I think this point IS important: You should not need to kick hard to get a horse to accelerate. If the horse knows the cue, and disregards it, then the solution is not to nag, but to provide a consequence that will motivate the horse to stop ignoring you.
There is a big difference between teaching a horse who doesn't know a cue what the cue is, and teaching a horse who knows the cue but ignores it. With the first, patience is needed, and slowly increasing pressure until the horse figures out what you want. That can take a long time with some horses. The rider should take as many weeks as needed...
But once a horse knows, it is refusing. That needs a consequence. I took group lessons for a few months some years back, and some of the horses could walk all day while I kicked as hard as my 180 lbs could. They had no doubt about what I was asking, but didn't feel like it. Thus the crop - and the lesson horses all only needed one hard whack to then move off a light squeeze.
FWIW, I normally hit MY leg first. The sound is usually enough. But since I ride my own horses, they know what to expect and I normally have to hit myself about twice a year...
Good luck. Also - talk to your instructor before or after about it. You are paying for instruction, and she knows her horses. She may also see something no one here can see, since none of us are there.