The self can be understood in objective metaphysical terms as a bundle of properties, as a substance, or as some other kind of entity on our metaphysical list of what there is. Such an approach explores the metaphysical nature of the self when regarded from a suitably impersonal, ontological perspective. It explores the nature and structure of the self in objective reality, that is, the nature and structure of the self from without. This is the objective self.
I am taking a different approach. In addition to objective reality, which is usually understood and explored from an impersonal, quasi-observational and metaphysically realist perspective, we can also explore the nature and structure of subjective reality. The nature and structure of subjective reality is defined by the nature and structure of first-personal, conscious experience. Subjective reality is as real as objective reality, and a metaphysical realist such as myself can endorse the existence of both kinds of ontology. The mental states that, as experienced from the first-personal or subjective perspective, capture the nature and structure of subjective reality, are included in objective reality. The questions to explore in a subjective ontology of the self concern the nature and structure of the self from the first-personal or subjective perspective, that is, the nature and structure of the self from within. This is the subjective self.