That's why I always go back to basics and never believe anything the customer says.
The figures provided at the start did not make any sense.
Simply connect the AC/DC wires to the battery - in either direction or if a red wire is provided, connect this wire to the positive terminal of the battery and the other wire to negative.
Connect the output of the sensor, possibly yellow, to one of the coil terminals on a 3amp relay and connect the other coil terminal to the negative of your battery. The 3 amp relay will be a 12v type with a coil resistance of at least 120 ohms.
When you push the button, the relay will click and when you push it again the relay will release. Make sure you get this type of switch.
Now connect the output terminals of the relay to the LEDs, by connecting one terminal to the negative of the battery and the other terminal to the black wire of your LEDs. The red wire goes to the positive of your battery.