Since I got my chocobo finally to rank 20, I can tell you this: You get all three bardings when you fill in all of the skill trees. Yes, you get enough skill points to do that as well. But since back in the days when I got my first barding, it dropped directly into my inventory as a usable item (and I don't have any memories of the second barding), the third barding set appeared directly (without any indication of doing so) into the barding list instead. And that's what confused me with all the skills and stuffs, hi ninja changes!