Does MoCA work if we use our own modem?

Well the good news is it looks like decent coaxial. Can't be sure if it's RG6 without seeing the writing on the coaxial wiring but it doesn't look too old and has good compression connectors. You should be able to use that.

The bad news is I had a thought about what I told you last night. Basically you can't have a separate loop for your MoCA because the MoCA adapter and modem have to be running off the same outlet. I also see two coaxial coming out of that splitter. I assume one goes to the room the modem is in and one goes to the phone modem? Is that right? If so, is the phone modem in a room where you game and want the 2nd MoCA adapter for the AP to connect to? If so, you at least know what coaxial you want to connect to the new MoCA splitter. If not, you need some way of tracking which coaxial goes where. You could do it through trial and error by moving the modem into where you want the 2nd MoCA adapter and try connecting different coaxial to the splitter until the modem comes online. Another way to test it is with a cable tester. Basically you connect one piece to the outlet you want to know where it goes, which produces a signal in the line, then use the other piece to attach coaxial wire to until you see that signal in the tool. Once you know which one is which, you can set up the coaxial by using the picture above. Don't forget the MoCA splitters. You should be able to reuse the MoCA filter.