Is this book's definition of phrase correct?
To be a phrase is to be reducible to a single non-terminal. What you call a definition of phrase is actually the definition of what it is to be a phrase of a sentential form.

$\beta$ is a phrase of $\alpha_{1}A\alpha_{2}$, because it will be reduced to $A$. There are (obviously) phrases in sentential forms consisting entirely of terminals, i.e., sequences of symbols that will be reduced to a single non-terminal. They are not phrases of that sentential form, though.

A handle is the simple phrase of the sentential form that will be the result of the reduction to be made immediately, at some point. $\beta$ exists as a phrase, because it will be reduced to $A$. A bottom-up parser will usually reduce first its leftmost simple phrase, which will then be the handle of the resulting sentential form.

Definitions are not arbitrary, they serve a purpose.