Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a clarification of the situation.
On page 25 of Gilbarg/Trudinger during the exposition of the Perron Method, barriers are introduced. The authors state that the existence of a barrier (i.e., regularity) is a local condition, and to support this claim they show how to construct an arbitrary barrier from a local barrier.
It is during this construction that I had a notational misunderstanding. At the end of the fourth paragraph, during the construction of $\bar{w}(x)$, they define $\displaystyle m = \underset{N-B}{\inf} w$ and make the crucial statement $m > 0$.
From my understanding, it should in fact be $m = \underset{\Omega \cap \left(N-B\right)}{\inf} w$
The simple reason - $w$ is only defined on $\Omega$. For instance, we can let $w$ tend to any negative number outside of $\Omega$ (but in $n$) to force $m < 0$.
After making the change, $\begin{align*}m &= \underset{\Omega \cap \left(N-B\right)}{\inf} w\\ &= \underset{\overline{\Omega \cap \left(N-B\right)}}{\inf} w\\ &= \underset{\overline{\Omega \cap \left(N-B\right)}}{\min} w\\ \end{align*}$ The last line follows by compactness.
Suppose $x\in \overline{\Omega \cap \left(N-B\right) } \subset \overline{\Omega \cap N} - \{\xi\}$. Then $w(x) > 0$, yielding $m > 0$.
So now my question: is my "counterexample" for $w$ given above correct? Am I missing a condition that the authors are using which makes their statement correct? And, most importantly, is my "hot fix" valid (replacing $N - B$ with $\Omega \cap (N-B)$? Again, I am new to PDEs, and the exposition of Gilbarg/Trudinger