Is this book's definition of phrase correct? I am fairly confident that my textbook (Concepts of Programming Languages 11th
edition, international edition, Robert Sebesta) has an error in its definition
of what a phrase is in the section on bottom-up parsers. After an infuriating
conversation with the professor using that textbook in which I was told that the
book's definition was correct, but was given no reason why or why my reasoning
was incorrect, I still think it has an error. I haven't been able to find any
information about errata in the book online (not the main point of the question,
but I'd like to find that also).

Here is the book's definition:

$\beta$ is a phrase of the right sentential form $\gamma$ if and only if $S \Rightarrow^{*}\gamma = \alpha_1A\alpha_2\Rightarrow^{+}\alpha_1\beta\alpha_2$.

Here is what I think the definition should be:

$\beta$ is a phrase of the right sentential form $\gamma$ if and only if $S \Rightarrow^{*}\alpha_1A\alpha_2\Rightarrow^{+}\alpha_1\beta\alpha_2 = \gamma$

Here is my reasoning: Definitions are mostly arbitrary, so I can't really prove
that it's "wrong" without comparing it with a bunch of other books out there (I
just have the one, so if those of you with more could check that for me, that'd
be great. I can't seem to find any definitions online). But I can prove that
it's not consistent with the rest of the book.

The book states that simple phrases are a subset of phrases, that the handle of
any rightmost sentential form is its leftmost simple phrase, and that in this
example grammar

$\begin{align}
E &amp;\to E + T \mid T\\
T &amp;\to T * F \mid F\\
F &amp;\to (E) \mid id
\end{align}
$

the handle of the sentence "id + id * id" is the first id. Observe that
according to the book's definition of phrase, a sentential form consisting
entirely of terminals can have no phrases, because the sentential form is never
of the form $\alpha_1{}A\alpha_2$. This means that there are also no simple
phrases of sentential forms consisting entirely of terminals. This means that
there are also no handles of sentential forms consisting entirely of
terminals. This means that there are no handles in "id + id * id", despite them
saying there are. So the definition isn't consistent with what else they
say. Based on what I've read elsewhere and heard in class, the definition seems
wrong, not the other parts.

Is my reasoning sound? Is the definition of phrase correct / what is the "correct" definition of phrase?

(apologies if I left out anything essential, I lost the first draft when I got to the captcha requirement and filled it out with spyblock enabled)
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.