Think about it people.. like.. hm..
Anime openings. The band makes the short version [which at times sounds different.. i.e: Inuyasha's Grip! from ELT.] and then they refine/complete the song in later stages. It seems to be the reason why singles sometimes come out so much later for anime/game OP/EDs.
But things really aren't the other way around since they'd have to add/subtract elements from a song, and usually CMs have the simpler version [if the CM is done quite a time before a release] It happened before up until like..STEP you. Nowadays the CMs aren't as far from the actual song releases..

I wonder if we'll get a 'CM version' collection soon... You never know with Avex!