I would certainly agree with this statement. However, I would note that it only has meaning due to the colloquial meaning of these phrases.
In colloquial American English, “living the good life” implies living a life that is full of material possessions and of what many people would characterize as “fun.” In other words, it implies living a life that is at least somewhat hedonistically pleasurable. People who live the “good life” do things like going to clubs. They buy many things that they can use to have fun. This is what we commonly call living the good life.
By contrast, when we speak of living a good life, we mean something altogether different. We are referring here to a life that is meaningful in a way that is deeper than “the good life.” We are referring to living a life where we build a good family and enjoy our time with them. We are referring to living a life in which we help others rather than thinking only about our own material pleasures.
Thus, these two phrases refer to very different things. On the one hand people who live “the good life” think mainly of themselves and of material things. On the other, people who life “a good life” think of others and of things that are more spiritual than they are material.