This book gives definition 2.7 wrt a vertex. It says:
Let $P$ be a polyhedron. A vector $x \in P$ is a vertex of P if there exists some $c$ such taht $c^Tx < c^Ty$ for all $y$ satisfying $y \in P$ and $y \ne x$.
This implies that a vertex could be a point between to extreme points. However, the text shows a figure, Figure 2.5, which states that $w$ is not a vertex. There seems to be a conflict here.
To make things more concrete, here is an example:
Suppose $P$ is defined as $x_1 + x_2 = 1$ subject to $x > 0$. I can make $c = [1, 1]$ and then make $x=[0.5, 0.5]$ which is not an extreme point, but still satisfies definition 2.7. What am I missing here?
The definition has a strict inequality. Your counterexample has a weak one.