Is the Scalar Product Definition in my book wrong?

Without context, it is difficult to say if there may have been a mistake. However, if done correctly, one of the other properties of $(\cdot, \cdot)$ will be the fact that $(\phi, \psi) = \overline{(\psi,\phi)}$, where the line denotes complex conjugation. In particular, you have that $(\phi,\phi) = \overline{(\phi,\phi)}$ and thus $(\phi, \phi)$ is always real.

Alternatively, for complex numbers $\alpha$, you could interpret $\alpha \geq 0$ as meaning "$\alpha$ is real, and furthermore $\alpha \geq 0$".

It is correct, because the only elements of $\mathbb C$ for which the relation $\geq$ is defined are real numbers. Therefore, the statement $$z\in\mathbb C\land z \geq 0$$ is equivalent to the statement $$z\in\mathbb R \land z\geq 0.$$

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We do not implicitly assume that the mapping is real. We only assume that $(\phi, \phi)$ is real for all values of $\phi$, not that $(\phi, \psi)$ is real for all values of $\phi,\psi$.

For example, the mapping $$(.,.):\mathbb C^2\to \mathbb C\\ (z,w)\mapsto z\cdot \overline w$$

is a scalar product, and $(z,z)$ is always real, however $(i,1)$ is not real.