OK thanks. In that case the chart is very wrong. They even elaborate on this very case, apparently incorrectly:
"For example, in Row 3, the left side of the table shows that TiVo Mini can connect to the host DVR on the local network, but has no independent connection to the Internet. The host DVR can connect to the Internet. Moving to the right, you see that without an independent Internet connection, TiVo Mini cannot use Internet-based apps or display poster art, but it can still access everything on the host DVR, as well as use Search, Browse, and the Discovery Bar."
I get that the mini can connect via moca to the roamio and that connection is bridged to the roamio's ethernet connection. In fact, that's what I'm relying on, since in my case, my mini won't have an "independent connection to the internet", which to me reads as ethernet only, not moca to ethernet via the roamio. Since obviously if the mini can connect to the roamio, and the roamio can connect to the internet, then the mini can potentially access the internet, though not "independently".