From the mere fact that they can't picture from the inside what it's like to have fainted and not yet woken up, it doesn't mean that nobody believes that they ever faint.
So, if you're woken up during slow-wave sleep you're going to be thinking "Did I take the-- yes, while I was dreaming I was thinking about the garbage," but if you're woken up during REM, "But a monkey was eating my grandmother," and that sort of thing. So, there is a distinction.