From what I understand, it is largely based on HER training...how SHE was taught the canter "cue"...
With my trainer's horses, the "cue" is as follows:
Upward transition from a working trot...
1) sit up and deep--not hard--just ensure full contact with your bottom so the horse feels you there;
2) half-halt to ensure the horse knows to be paying attention as you are preparing to ask for a change;
3) apply outside leg AT the girth, while ensuring no inside leg PRESSURE, (this is perhaps where you two are having confusion together-- if you ask with both legs equally at the girth, the cue you are giving is simply "forward", or " faster", and not a change of gait).
4) Give slightly with the outside rein (same side you're applying leg) and apply slight balancing tightness with the inside rein; this gives horse some rein pressure to balance against, especially if new to cantering on a circle/in an arena. As well, between the outside leg and pressure and inside rein taughtness(slight only), this contributes to the desire in the horse to pick up the correct, or inside lead, as you are driving her forward and slightly IN through the application of ONLY outside leg @ girth (either a squeeze or a tap-tap, based upon how sensitive your horse's sides are). The"slight pushing over" effect with the outside leg drives horse to take the initial canter step with the INSIDE foreleg, thus putting horse on correct lead if they are not inclined to pick it up automatically.
5) If you need additional "steam", utilize a cluck/kissing noise AS you apply the single leg pressure AND possibly try verbally cueing with an encouraging
and firm sounding, "Canter!"
6) IF your horse doesn't pick up the canter in the first one it two steps, don't continue cueing as this will probably just drive her into a more forward trot, telling you she doesn't "get" what you are asking (language incompatibility!) & I am told to then transition downward right away to the walk, picking up a "moving on" walk, nothing "pokey", and FROM THE WALK, NOW, ask in the very same manner as above for the canter.
7) If she doesn't pick it up after multiple, clearly and well communicated attempts, (always transitioning downward if you don't acheive canter within one or two steps max, & beginning again, so as not to further confuse her re: what you want), make sure to also then try a change of direction and try from the other side...perhaps your horse was often cantered to the left/right, so didn't integrate the cues properly from both directions? That's one way to find out. Also she could be less likely to pick up the canter and correct lead on her "stiffer" side, thus better INITIAL RESULTS going the other direction?
Finally, horses of varying disciplines or with less formal training pick up canter with a variety of different cues as I was told. Thus if your's is an OTTB, perhaps she had a whole DIFFERENT BASE OF TRAINING than my trainer's dressage and h/j horses!!
**This is where someone with a lot more knowledge than myself needs to step in and help out! I do, however, sincerely hope that something I've said may be useful to you...when the trainers and experts jump on the thread, my guess is we'll BOTH LEARN SOMETHING!
Best to you!! B2H