Where does then belong identity into this? Oftentimes we see personal identity being discussed as a part of the self, but more of a form of its representation.

Personal identity is stating the facts about our core - the self. If thoughts are the defining characteristic of cognition, identity is then a defining characteristic of the self. In comparison to the self which has almost a metaphysical presence at the center of a human being, personal identity is the acting of the self in a form of a physical and subjective phenomenon