Thanks guys! Regarding the connection suggestions:
When I disconnect the direct-to-TV HDMI cable, and connect the Cox Cisco box straight to the TV with a coaxial cable (which is one thing I hadn't tried yet) and change the TV's input source to "TV", I get a sort of rolling snow picture with the small "No Signal" message floating around. Changing the TV's input signal setting to the other options does no good.
Not being a techno guy, the "RCA/Composite" term has little meaning to me, but I assumed that meant the assortment of cables with the red/blue/green-colored connectors combined with the white/red combo, along with the other option of the white/red/yellow trio. When I disconnected the HDMI cable and connected that whole red/blue/green & red/white color connector pile directly to the TV and reset the input signal source to "Component (DVD)" ….. voilà, the cable box signal came in just fine even though it actually bypassed the Magnavox VHS/DVD machine completely. The temporary little gray square on the TV screen says 1080i at the bottom. When I did this same direct Cisco-box-to-TV connection for the 'lesser quality' white/red/yellow trio, it also showed the TV signal just fine, and the temporary little gray square on the TV screen says "NTSC" at the bottom instead. Whatever that means.
So next, I tried connecting the old white/red/yellow cable trio from the Cisco box to the Magnavox and then on to the TV with a second identical trio set, no result. (Although if I put in a DVD, I can watch it, of course). Next, I left the old white/red & yellow cable trio from the Cisco box to the Magnavox alone, but connected the green/blue/red video trio & red/white audio pair from the Magnavox to the TV, same no signal result. After that, I left that last Magnavox-to-TV connection alone, but tried connecting the coaxial cable Cisco-to-Magnavox connection. No signal, still.
Next, I tried the coaxial cable Cisco-to-Magnavox connection but with the HDMI Magnavox-to-TV cable hooked up (I'd mistaken the Magnavox's HDMI connector as an "in" connector, when it is only an outgoing one). Only a blank gray screen results, which later defaults to a power-saving screen with the Magnavox logo floating around. In all of these, I tried it with the Magnavox turned on, and turned off, where the input sources on the TV were "Component (DVD) when it was on, and "TV" when it was off. That's the way my prior Scientific Atlanta cable box worked.
And as I noted at the start above, the Cisco-straight-to-TV coaxial connection showed no signal. I would have thought that would work like the color component cables did, and like the HDMI cable did when they were each connected directly to the TV.
So, at a minimum, I can say that a TV signal is being sent out of the Cisco box via both the HDMI connector and the color-coded connectors, but I don't know about the outgoing coaxial cable connector.
The ultimate goal is to still see a TV signal with the Magnavox recorder on, so that I know I can record onto DVD what I am seeing. Old as that machine is, it is still occasionally handy for dubbing old VHS tapes onto DVDs, so that those old recordings can be watched at other people's homes who only have DVD players.
