Both circuits you posted have a direct short to ground! DO NOT USE!
First, you want to power the relay coil off the headlight wire: Lights off, relay off; light on relay on. Ground the other end of the relay coil (the unlabled pins on your relay picture).
Second, you want to find a spare circuit in the fuse panel which has 12V out of it only if the key-switch is in the run position.
Install the appropriate fuse for your LED lights, and hook that to terminal 5 of your relay. Hook terminal 1 to the + input of your LED lights with the - end grounded to airframe.