As a mathematician, I agree. But when you talk with people that work in the applications, they usually have weird definitions.

In particular, in this book they don't use $Z_2$ that is the usual notation in math (at least here). They use $C_2$ that is more usual in physics. And for the selection of the groups I guess the author has some interest in crystallography.

Yes the motivation was crystallography and giving a small amount of group theory background was about providing the groundwork for talking about it.