Canonically, you are playing "the" Guardian's story, but is purposely never stated the race or class of the Guardian. Several stories do this, so that it is written with the player as the hero, everyone else being just another Guardian in your fireteam or in the Tower. So technically it is a weird mix of all three races and classes, kind of like multiple versions of a myth, which means your own head-canon is correct and as valid as the next person's. (Personally, my own head-canon is my Guardian is just re-living the campaign or other activity through a simulation of their recorded events.)