How do you find the inverse of an arbitrary $f(x)$ if $f$ isn't one-way?
Both (1) and (2) are possible.  (It's also possible to have a combination of them.)  It doesn't really matter which holds; either way, that means that $f$ is not one-way.

You say (2) is false because of the preceding discussion, but I'm not exactly sure why you think it is false.   One-wayness is not a property of a specific value of $x$; rather, it is a property of a function $f$.  And, just because a function $f$ is not one-way doesn't mean it can be inverted for every possible output.  If $f$ can be inverted on a significant fraction of outputs, it is not one-way.  If $f$ can be inverted on all outputs, with a significant probability, it is not one-way.

Perhaps the confusion comes from a misunderstanding about what the definition of a one-way function is.  The definition you quote at the beginning is not the definition of a one-way function.  Rather, it's the definition of a hardcore bit.

The part you quoted appears to claim/imply that it is the definition of a one-way function with hardcore bit (which is not the same as the definition of a one-way function).  That claim/implication is not correct.  The precise statement is that this is the definition of a function with hardcore bit.  This definition implies nothing about onewayness.  So why did they call it the definition of a one-way function with hardcore bit? I don't know; you'd have to ask the author. But my guess is that this definition is normally only used with functions you already know to be one-way, so there's an implicit assumption that $f$ is already known to be one-way, and given such an $f$, the definition is describing what it means for $b$ to be hardcore for $f$.

You say "suppose $f$ isn't one-way, then this expectation will be noticeably different from 1/2".  That actually doesn't follow.  You can have a function $f$ that isn't one-way but does have a hardcore bit.  Then the expectation will not be noticeably different from 1/2.  E.g., consider the function $f$ given by $f(x)=0$: it's not one-way, but it does have a hardcore bit, and the expectation is exactly 1/2.

It sounds like you want to understand one-wayness.  If so, hardcore bits are a distraction.  You should be looking for the definition of a one-way function and trying to understand that, not looking at the definition of a hardcore bit.