Limited dragons are most certainly not included, I got the achievement without having the Bone dragon, and I also didn't have a Plant dragon at the time either, so I think it only tracks whether you've ever had the dragons, despite the description.
There's lots of speculation on the Wiki and it seems like some people just aren't getting the achievement, but it might be due to game center being laggy or something.
It appears to be the case that you need all non-limited dragons, including rare dragons like the Rainbow Dragon. This doesn't include limited dragons that are currently available; only year-round dragons count, but you need all that are currently available. It seems this is a moving goalpost, meaning as new year-round dragons are added, those are added to the requirement.