Using the Fey Creature template on an animal turns it into a fey, not a magical beast. IMO, the fairy beast is some mind of fey creature, rater than an awakened animal.
I'm not really doing more work on this class presently, but just to clarify this question: I think of the faerie beast as a creature akin to a fey creature (i.e. a creature with the fey creature template). It is some sort of fantastical animal-like creature from the world of the fey. In my mind a player could use the critter's evolutions to create a completely new kind of fey creature (a horned lynx, a mantis centaur, etc.) or replicate some existing creature with connections to the world of the fey (say a faerie dragon or similiar). Obviously, in the latter case you would not be able to wholly replicate any specific monster's exact abilities, so it'd have to be a kelpie or whatever with unique and strange abilities. Personally I prefer to make up unique creatures; the First World is vast and ever-changing, after all, and could potentially hold all manner of unique and rare creatures as yet undiscovered.
EDIT: I guess a faerie beast is really roughly as much "its own thing" as an eidolon is. I mean, some people prefer to play a summoner who summons an actual, unique angel (or demon, or protean, or whatever), whose abilities have been magically altered via their special bond, while others prefer to think of "eidolons" as a specific creature type and say their summoner is only summoning an outsider of the type "eidolon" who mimics the traits of an angel. I never really cared for the latter approach, but that does seem to be how Paizo wanted eidolons to be treated. Robs summoners of a lot of flavor opportunities in my book, but that's another discussion; one I'll be looking for views on on the Advice board.