There is nothing wrong with counter cantering and I totally agree with JustDressageIt. "Corporal" (1982-2009, RIP) preferred his left lead ALL OF HIS 23 YEARS WITH ME. He would pick up the right lead when I insisted but I ASSURE YOU it wasn't a pain issue bc that horse would go all day without any lameness.
I agree that starting and stopping is confusing. You might want to go back to the lunge while he's saddled and teach him the English for "right lead" and "left lead". It will translate when you are riding.
I've also watched Julie Goodnight's many programs that involve correct leads. She can get any horse to pick up the correct lead immediately by positioning the front foot forward, haunches at a 45 degree angle.