Communism IS socialism.
Well, yes and no. From a liberty perspective, they are the same. However, Karl Marx said that socialism was an "intermediate step" on the way to full-on communism. So I was refuting the authors' definition with one of my own.
As I see it, all forms of socialism have in common the idea of "Control of the means of production in the hands of the state." There are at least three ways to do this: 1) Ownership (communism), 2) Control via regulation (European style or "democratic" socialism) or 3) Partial ownership with the state as the majority stockholder (Fascism).
What's the difference? None, really, especially when we realize that the "means of production" is us!