I believe that you will find that your A2 "birdcage" is a true flashider, and not a compensator. ATF's classification is correct! The muzzle end opening of the A2 is of full diameter, just like that of the A1. That is part of how a flashider works.
The closed-off bottom slot of the A2 is rumored to have been first implemented by the Canadians as a response to the A1's open bottom causing snow to fly, obscuring sight, during prone firing. It is said that the US adopted this design for the M16A2 after realizing that it worked just as well on sand!
True, the diversion of muzzle gasses in the A2 design somewhat reduces muzzle rise, but don't you think that a better compensator would have been implemented if that was the objective? A true compensator will have the muzzle opening diameter reduced to that of bore diameter, or maybe a little larger, in order to properly perform its function!