The mind doesn't usually "shut down" in the relaxation response, although having arrived there, in a physiological state different from sleep, someone might pass through it and nod off.
People do, or experience without "doing," all sorts of mental states once they reach the relaxtion response.
Some people prefer restricted and patterned activity to "sitting down." Similarly, the "shutting up" can take the form of restricted and patterned thought, such as mantras or mindfulness.
And some people just look at their stamp collection. But not everybody who looks at their stamp collection is meditating. Then again, not everybody who sits down mute is meditating, either.
Meditation is something like the intentional stepping away from ordinary involvement and concerns, while retaining wakeful consciousness. If it is done at all, then it is done correctly. The physiology, left to work for itself, works by itself.