It appears that the diagram posted is for a bus with separate stop and turn lights and a towed vehicle with combination stop/turn lights. Wouldn't a different arrangement be needed for a towed vehicle with combination stop/turn lights?
Typically, even if a vehicle has separate stop and turn lights, which most new vehicles do, we only use the the stop light for both stop and turn. It doesn't matter if you blink a separate yellow light for turn or blink the red stop light for turn. Jack's schematic will provide the blinking stop light for turn signal. My schematic actually has the ability to drive a turn signal which is separate, though I know of nobody, including me, that has done that. Most trailers you buy will have a combined stop/turn signal, so if you want to be able to pull things other than just your toad, you want to wire the toad for the most general purpose, which is combined stop/turn.
Wouldn't it be better to also have a ground wire between the bus and towed vehicle so you wouldn't depend on the hitch for ground?
Absolutely. You should always run a ground wire through your plug and harness. It's just not shown on the diagrams.
Are all cube relays configured the same? It would be nice if the terminals in the drawing were numbered so novices, like me, would know how to attach the wires correctly.
See Jack's comments above. I agree with what he said. I think the cube relays are standard, but not 100% sure. Usually, they have a diagram molded on the relay which shows
the terminal numbers and how they work.
Our bus has separate turn and stop lights and our car has combination stop/turn lights. There is a relay system in our bus now but the turn signals don't work on the car. The tail and stop lights do. I think the PO had the relays set up for a car or trailer with separate stop and turn lights. There are five wires used on the receptacle on the bus and only four wires used on the plug of our car. I thought I would need a converter between the bus and the car but now I think if I change the wiring to match the posted drawing the turn signals will work. If my thoughts are wrong please correct me.
I expect if the PO already had relays in there, you probably just need to modify the wiring on it to match what Jack's shows, and you'll be good. You may also have to rewire the hitch plug, too.
If we get together this spring, I would be happy to help you get this fixed.