Does a Vertex Cover exist? This should be a simple question, but I am a little bit confused.

A proof on page 556 of Algorithm Design says:

"Let $e=(u, v)$ be any edge of $G$. The graph $G$ has a vertex cover of size at most $k$ if and only if at least one of the graphs $G-\{u\}$ and $G-\{v\}$ has a vertex cover of size at most $k-1$."

But when I try the following example where the black ball shows a vertex cover, it seems to be faulty statement. Because $k$ remains as $k$ (not $k-1$) even after deletion of vertex $u$. Moreover, if I delete the vertex $v$, there would be no vertex cover left.

What is wrong about my deduction and example?
If you delete the vertex $v$, then there are no edges left at all, and so there is a vertex cover of size $0 \leq k-1$.

Why is the statement true in the first place? Suppose that $G$ has a vertex cover $C$ of size $k$. Since $(u,v)$ is an edge, $C$ must contain at least one of $u,v$, say $v$. I claim that $C \setminus v$ is a vertex cover of $G \setminus v$, from which the statement immediately follows. Indeed, let $(a,b)$ be any edge of $G \setminus v$. Then $(a,b) \in G$, and so $C$ contains one of $a,b$, say $a$. By construction, $a \neq v$, and so $a \in C \setminus v$.