Counter Cantering is wrong unless being asked for specifically :) In judges command classes they will often ask for it to see if you have that kind of control and communication with your horse. If you can specifically ask for a counter canter, you can certainly ask for the correct lead, you can easily change between leads during a riding pattern that requires direction changes, you can work at getting flying lead changes down.
To find the lead you are on, I was bad for a long time...I leaned to the direction of the lead I was checking for and looked down at that leg to see if it was jetting out. I cannot begin to express how bad of a habit this is, as it throws your horse off balance, and looks tacky. A more subtle way could be to watch her shoulder, it will have a more pronounced movement than the other. That also is not most desirable. The best way is to learn the feel. It will take a lot of time and practice, and maybe even someone's help. You can call out what lead you think you feel, and a helper can tell you if you are correct or not. The more you practice picking up the different leads, the better you will get at feeling which lead you are on.
Try to work on picking your leads up on a straight stretch to keep her from learning to pick up the lead in the direction of the bend, but instead to pick up the lead that your legs are asking for.
You of course want to keep working on balance at the slower speeds, but also practicing with your leads in saddle will help her learn to balance over time.
One thing to remember is that it is her hip that drives the lead, and the shoulder follows. It is possible for the shoulder to pick up the correct lead and for the hip to be on the wrong lead (feels horrible). Since the hip actually drives the lead, you want to work with her ability to move her hip over when you ask with your legs/reins, which will in turn help with your leads, h.alf pass, etc.
Do you know how to do hips in?
I hated learning it, but now that dez has it down (much room for improvement...I want to be able to trot and canter with hips in), he picks up his leads very easily. Our communication is much clearer.
I worked on teaching hips in, and then went to cueing for a particular lead at canter...I put my inside leg on girth, and outside behind. I press with my outside leg saying move the hip in...then squeeze with my inside leg to cue the gait change....it works really well.