Unfortunately, a lot of songs are mislabeled. Particularly on filesharing software, you'll see a lot of Ayu songs that don't really exist, or existing songs that are incorrectly labeled.
This is how I managed to believe that the Eurosenti mix of Depend on You was the original mix.
I believe the only anime she has been officially involved with to date is Inuyasha (Dearest for the third ending theme; no more words for the movie) and the game Thousand Arms.