IMHO, moving the center is perhaps some of what they have been talking about in the internal aspects, its very subtle and perhaps isolated, standing.
Moving from the center is to initiate full body movement (irimi, tenkan) from the center.
Actually my experience has been the opposite, that the internal stuff de-emphasizes "moving from the center" at least in the sense that I have seen it in aikido. Rather than a strong emphasis on the hara as a focal point for initiating movement, there is an increased awareness of the rest of the body and longer arcs of tension/connection. It's been my experience that when you try to initiate movement *from* the center, the rest of the body lags behind slightly, but moving *through* the center can stabilize the body better. Subtle distinction, but there ya go.