Thanks for your help! I tried several of your suggestions, attached an external 5v power supply and changed the circuit to make that work, and in the end it turned out that not the data but the power connection was giving trouble. After connecting an elco of 470 µF to the Vcc and Gnd lines at the end of the cable, it all works. (I suppose 470 µF is a lot, but I had it available and it does its job.)
BTW the relays are actually an optocoupled set of two, but I took the first good-looking relay pictures I found in the layout program. (Sorry.)
After a week of struggling and misunderstanding, I can finally get back to programming the thing. Which I find much more fun. 