The light does not turn on because it is not powered. You have the ground connection to the switch, but the Battery connection of the switch is also connected to ground. The "output" of the switch is now ground when "on". You are doing what is called "lowside" switching, connecting the ground to a device to turn it on. This is normal in cars, but won't work with this type of lighted switch.
If you follow the "lighted switch and relay " diagram above, the switch will light.
For the light to work, it must be connected to both ground and 12 volts ( battery positive ). Your second diagram connects ground to both sides of the internal light, so does your last idea above.