My motivation for debating the two uses of the back button is whether I prefer a smooth OS integration (back as in back in time) or whether I prefer displaying more ads and am willing to capture the user inside (back as in up within my app).

It has unfortunately little to do with UX, since both end up being confusing on edge cases (Astro's horribly enraging "press back twice to 'exit'", the conversation or gmail app issues people mentioned in this thread).

I personally tend to prefer back as in back in time but I know for sure opinions vary.