Chucks site hasn't been updated in a while although it still has a lot of good information on it. The definition of "Aquatic Plant" seems to be broad enough to include plants that can live their life cycle completely submersed (including flowering) as well as those that live emersed or floating. I believe that the plants being sold as non-aquatic probably decay if submersed for too long (6 months or so?).
I've never heard of or experienced Alternanthera reineckii (rosefolia, lilacina or other variants) suffering decay due strictly to being submersed so as far as I can tell it should be considered a "true aquatic".