The charger is switched on when the switch is in the "up" position - ie the upper part of the switch is pushed flush with the mount. This is the 100V mains charger however, and has no charge input to the Cube unless it's connected to a 100V (Japanese mains) supply). Best to only try this when stationary

For the mains charger to work, the "mains" switch at the front of the Cube, under the fridge socket, needs to be in the "up" position. Afraid that none of this should have any bearing on your charging problem though.
From memory, I think that the charge controller light will come on when the engine is running, even with the battery disconnected. I think that the red LED cut-out switch will also be lit, but not sure on that. Is it possible that you have a break in the connection to the battery so that it's not actually connected to the Cube charging system? Breaks can occur - my loom wore through at the point where the =ve &-ve leads join the main loom, shorted out & set fire to the wiring!
I still suspect that the relay is at fault however. Symptoms sound very much the same as those I had with a dead battery. Holding down the cut-out switch would allow me to operate the electrics, but it wouldn't stay on. While the switch was depressed & any of the electrics operated I could hear a constant clicking from the relay as it tried to cut the circuit.