There are several solutions to the problem:
Use a 5...6 V Zener diode and a relay with 5..6 V coil. Adjust a series resistor in the Zener diode-relay coil circuit so that the relay operates at 13 V and its contact opens the battery circuit.
Use an adjustable comparator to open relay contact to battery at the selected voltage.
One-transistor relay driver can do the above function as well.
You can make precision voltage regulator allowing the 13 V output to the battery.