What exactly is the nature of this quasi-extraneous "self" that is not contained in the mere body that my consciousness observes, but which I must believe in and in fact know exists because I am having those observations.
That is a question I have been asking too.
As you say, at one moment the self can be observing something it percieves as outside of itself.
At other times it is observing itself. But it cannot really do that any more than the eye can see itself. To do it, it needs a mirror.
My take on the nature of self, on account of the implications of the above, boils down to this.
Self is merely a concept. An empty shell which we fill with meaning according to what we need in any given circumstance.
Self is the notion of an absolute, the dualistic counterpart of the absolute of "other" or non-self, also known as god.
But self changes and shifts. It is never constant, and it is impossible to tell where it's borders are. Where does the self end and the other begin?