How to allow only 12v
You might be able to find a 12v relay that has a pull-in and drop-out voltage greater then 5v.  That way a 5v signal will not turn it on or hold it on.

If such a relay can not be found you can create one.
Use a relay with a 5v coil.  Place a 6v zener diode in series with the relay coil, (not in the wire going to the LEDs).  The zener will need to be rated with the proper reverse current and wattage to withstand the on current of the relay's coil and the expected power dissipation, (Vz x I).

With the above setup a 5v signal will not be enough to force any appreciable current into the relay coil. At 10 or 12v you should get 4v to 6v across the relay coil.  You will need to check the relay spec's to be sure that voltage range is acceptable.  If the LED line voltage is consistently closer to 12 then you might try using a 6.8v zener.  

Note that zener voltage parameters often have a wide tolerance so you may need to do some testing to find the best part.  Also remember to place a reverse standard diode across the relay coil to reduce any flyback voltages from getting onto the line.