Should the vertex be constant because its slop is equal to zero?
Monotonicity is generally defined on a subset of the domain which includes multiple points e.g. an interval, so (again, in general) it doesn't make sense to refer to monotonicity at one single point.
(As a side note, it does make sense for differentiable functions to associate the derivative at a point with the "rate of change" in some sense, and it's common to say that $f(x)$ is increasing at $x_0$ if $f'(x) \gt 0$ however that's more of a casual language license, and also it's a sufficient condition but not a necessary one, for example $f(x) = x^3$ is strictly increasing on $\mathbb{R}$ but $f'(0)=0$.)
Back to the question, what the book says is correct. However, it is equally correct, and in fact a stronger statement, to say that $f(x)$ is strictly decreasing on $(-\infty,0\,]$ and strictly increasing on $[\,0,\infty)$ where both intervals include $0$.
And how come I can leave it while it is from the domain of the function?
There is no requirement (or guarantee) that the intervals of monotonicity cover the entire domain.
For example $f(x) = x \cdot sin \frac{1}{x}$ for $x \ne 0$ with $f(0)=0$ is a continuous function which is not monotonic on any interval that includes $0$. For an example of a discontinuous function which is not monotonic on any interval consider the indicator function of $\mathbb{Q}$.
