"Authenticity" and "inauthenticity"

I suggested in my last post on Heidegger that the distinction Heidegger makes between "authentic" and "inauthentic" understanding is not related to genuineness essentially. To support this, I cite the following:

...Authentic understanding, no less than that which is inauthentic, can be either genuine or not genuine.
-Being and Time, I.5.31, 186

The implication is that genuineness is not the defining characteristic of authenticity: what is authentic is not necessarily genuine. Now, I also said that because genuineness is not part of the picture, this distinction authentic/inauthentic is also not related to a traditionally metaphysical conception of truth as correspondence or (fundamentally) mimesis. I'll elaborate this latter point some other time. The first, however, is just as important for setting up this elaboration. What does Heidegger mean by saying that the phenomenon of understanding can occur in these two ways? Where does the distinction originate?
It originates in two places, only one of which we'll discuss--though both of course as features of Dasein's Being are reflections of each other and two aspects of the same phenomenon (for simplification, however, we'll keep them seperate). The first, which we will not discuss, is in the type of temporality characteristic of Dasein. The second (which appears first in Being and Time) can be found in Heidegger's following remarks:

Why does the understanding... always press forward into possibilities? It is because the understanding has in itself the existential structure which we call projection. With equal primordiality the understanding projects Dasein's Being both upon its "for-the-sake-of-which" and upon significance as te worldhood of its current world... Understanding can devote itself primarily to the disclosedness of the world; that is, Dasein can, proximally and for the most part, understand itself in terms of its world. Or else, understanding throws itself into the "for-the-sake-of-which"; that is, Dasein exists as itself. Understanding is either authentic, arising out of one's own Self as such, or inauthentic.
-Being and Time I.5.31, 184-6.

Dasein either understands itself in terms of its Self (in terms of its own "for-the-sake-of-which," its own existentiality, its Being), or in terms of its world (in terms of significance). Authentic understanding is the first, inauthentic the latter. To this should be added the crucial sentence that follows these remarks: "The 'in-' of 'inauthentic' does not mean that Dasein cuts itself off from its Self and understands 'only' the world." That is, understanding "in terms of" the world (inauthentic understanding) does not mean that the world only is what a Dasein understands. It means Dasein transforms its existentiality into something with significance and not into the Self; or that Dasein transforms itself into something that is understood like the things within the world instead of seeking out that character that is not worldly but is distinctive for it (the chraracter of the Self as existence).
I'll elaborate all this later in a clearer fashion.