So I got fed up with troubleshooting the long chain of cables and connections, some of which I doubted were done right by previous owners.
Instead I traced the shielded boot connectors from the vehicle harness to the trailer light outlet inside the wheel well. Thinking the outlet might have its own problems I don't need to find first.
My magnetic light kit is four wires, and previously in this thread the Collective Wisdom identified which numbered tag goes where. I spliced male boots into an octopus four-wire harness (another story for another time) and in daylight with a cup of tea beside me began.
The very first attempt at plugging plugs into tag-identified female connectors was almost correct! Left, right, markers and brakes. But, both filaments lit up and the lights went off when the brake pedal was pushed. Hmmm? More jiggling and testing, and confirmation with the family vehicle that the light kit wasn't acting up. Finally, I tried three-wires plugged in and the white grounded to the body. Success in every switch setting. Great joy and satisfaction.
The only thing to fix before making a road test, is a previous owner built the trailer hitch receptacle by hand. My standard-sized light to medium duty hitch is too small by about 1/8" all around and the pin is too short. Bugger!
Off to Princess Auto to see the nice people who stock the shelves with stuff I just have to buy. Found a male/female receptacle adaptor, a longer hitch pin and a better ball. Double bugger!!
That same previous owner made the receptacle just a bit too small for the adaptor. Time to consider the options in daylight with another cup of tea.