I bet if the OP check's this out there is no usable power on the line.
The gate's we install we use cat 5e it using 4 wire from it one set c the gate opens second set closes the gate.
But maybe his gate is not made the same but the plug is RJ11 we run 5e tho because the computer uses it and just use 4 wires for the gate. saves having to keep both boxes of wire.
These gate motors are from China also and they also call it a dry contact. But the computer function is a input switch function with current limiting. the line can maybe sink 10 mA to ground.
You have to find power somewhere else