Electronic – XOR in electronics

From the picture it looks like the relay you're using is an Omron G5LE-1. The wiring diagram is available here.

It's a SPDT relay, which means that there is one switch element that moves back and forth between two contacts, unlike the SPST relay used in your circuit diagram. You can still use your relay, but you'll have to wire it so that the "throw" part of the switch is connected to the unused "pole" when the relay is de-energized, and to the other pole when it's energized. From the manufacturers diagram, figure H, it looks like you would want to connect pins 3 and 1 to your output load, leaving pin 4 disconnected. One of pins 2 or 5 would be connected to the collector of the transistor, and the other to the power supply.

You can build the mux with 4 NOR gates, too. It doesn't require extra inverters on the inputs and outputs, if that's what you were doing. Note that the sense of the "select" input is inverted, so you need to swap the "A" and "B" inputs to compensate.