To access the TV or Internet signals you need a MOCA box that picks up the type of signal you want (TV or Internet) and puts it onto the right kind of output (coax or HDMI or RCA plugs for TV, or RJ45 for Internet.) Your TV STB is a MOCA device. The coax cable connecting your TV to the STB carries only the TV frequencies; you cannot use that cable for INternet stuff.
The router talks to the STBs via ip. Nothing to stop you putting a splitter before the stb and and adding another moca device which will get it's own dhcp address from the router.
I think your post is contradictory. You said "Nothing to stop you putting a splitter before the stb and and adding another moca device" which I totally agree with - the coax going in to the STB comes from the ONT and has both Internet & TV signals on it. So any splitter you put on the coax before the STB will produce 2 (or more) coax cables with the same set of signals.
What I sad was the coax coming out of the back fo the STB has only TV signals on it (actually decrypted ones since the TV signals sent by FIOS are encrypted, and one of the jobs the STB does is to decrypt them) - so only TV sets or something capable of receiving TV signals can use the STP output coax.
Yes, the router does talk to the STB via IP, but my understanding is the STB simply intercepts this and performs the commands sent via IP. It also sends responses via IP back to the router - which is why you can order PPV movies etc.. But I have never seen any evidence indicating the STB puts any sort of IP onto it's output coax.
Ah yes, when I read it again I see you refer to the stb to tv connection. It never occured to me that anybody would consider that for internet connectivity. As you, and he, said putting the splitter right at the ont and connecting his old coax network to one of the connectors should accomplish what he is trying to do.