Definition of a distinguisher
I think of it this way: think of a distinguisher as an adversarially-chosen statistical experiment that attempts to support or refute some hypothesis, that again is adversarially selected.  This means that the honest party doesn't get to assume what meaning the adversary assigns to $0$ or $1$; their PRG has to behave like a random distribution no matter what the adversary does (with the proviso that the adversary's experiment must complete within polynomial time).

For example, a distinguisher could be an algorithm that takes the output of the PRG, chooses a random position in the bit string, tests whether the bits starting from this location are equal to some ASCII representation of the Gettysburg Address, and outputs $0$ if they are, $1$ otherwise.  If that's the statistical test the adversary wants to use, they can do it.