Arduino Uno - Multiple modules for HDMI switch device
The relay wiring diagram you are following is just plain bizarre and the strangest one I have ever seen.

Ignore it. It is completely incorrect.

You only need one pin to control a relay, so disconnect it from pin 13. 

The switched side can connect to anywhere where you would normally use a switch - such as to switch power to an LED connected between +5V and GND with a suitable current limiting resistor.

You can use any available pin to control the relay, and that includes pins 0 or 1 if you aren't using serial communication.  Note that with the relay connected to one of those pins it will most likely activate and buzz while you upload a sketch - best to disconnect it at that point and reconnect it afterwards.



Devices like relays and LEDs need their own dedicated pin. They can't be shared with other functions or each other.

The only devices that can share pins are devices which connect to a bus, such as SPI or I2C.