Likely Solved
Of the options people have presented, a video card is by far the most likely for us to have owned at the era those options are from. The two-way arrow symbol on the connectors does give a little bit of doubt, but it seems pretty clear at this point that if I still owned the matching product, I wouldn’t use it, and that’s enough info for my needs
Both din connectors have a two-way symbol 
It is definitely 7 pins, not 9. The pin layout matches this image on the wikipedia page for mini din: https://commons.wikimedia.org/wiki/File:S-Video_7-pin_quasi-DIN_connector.JPG#/media/File:S-Video_7-pin_quasi-DIN_connector.JPG
I tested the pin connections and it does match the “compatible with an S-video… plug” in the image’s caption, but I don’t know if the keying allows for that. The key on the 7-pin is both wider and thinner than on the 4-pin.
The fact that this is a long cable instead of a short adapter does give me the impression that both the 4-pin mini din and the rca connector are supposed to be used simultaneously.
I don’t believe we ever owned a capture card, but it’s at least plausible to me that an old video card may have used it.
Testing with the multimeter, the outer pins of the 7-pin connector are 1-to-1 with the pins in the same places of the 4-pin connector. I read the wikipedia article on mini din connectors more carefully, and there is an indication that this scheme was sometimes used to have a socket which could accept either an S-video cable or the proprietary one. However, the keys don’t look compatible. The key on the 7-pin is both wider and thinner than on the 4-pin.
the center pin of the row of 3 connects to the pin of the rca connector, and the ring of all 3 connectors are connected together. The center two pins of the row of 4 are not connected.
That is indeed a Nintendo RF converter in the background, but I don’t believe we ever owned a Sega console. Granted, I didn’t know we had a SNES until it was found a few years ago in a shed that was being torn down
I wasn’t aware that there were any games that could make meaningful use of 16 cores, let alone games that might want more. Was there a major advancement in game programming when I wasn’t looking? Or is the headline as far off base as I think?