The counters are indeed confusing. Being a mount collector myself, I watch them regularly hehehe.
Log completely out and in, that can sometimes trigger the achieve.
Log in to a toon that has extra class mounts, like a pally or a warlock.
Open your achievements. Click the “Statistics” tab, highlight “character” on the left and then scroll down to “gear.” There is a number for total mounts (usable by that toon) that I’ve found to be more accurate than the achievement counter. See if that number is 200+.
I had a friend trying to hit 300 the other day. He had a heck of a time getting his 300th mount to show up. After systematically logging many of his alts, and reloading the game a few times, the achieve popped up on a random login.
Here’s a few other sources you might check for info:
