I tried looking at criteria to determine progression, but the problem is that, depending on the achievement, criteria seems to get reset once you complete the achievement, or else if you completed the achievement prior to the great Account-Wide Achievement Conversion of 2012, the criteria never got set. For example, on this character I have achievement #1699 (Fistful of Love) complete, because I completed it back in 2010, but none of the criteria for that achievement are listed in my criteria list (6302, 6312, 6313, etc). Meanwhile, my alt who hasn't even begun the achievement also doesn't have any of the criteria. Another alt who has started the achievement has some of the criteria complete, so I know that char hasn't finished it, but in criteria there seems to be no difference between 'not even started but account completed it' and 'specific character has completed it'.
achievementsCompletedTimestamp doesn't seem to help, either; all of my character have the same time stamp for a given achievement, even though in-game it varies.
I'm getting the feeling that what I'm trying to find isn't actually exposed by the API, but am I missing something else?