Reading your post above makes me realize that the Hunter's way is to let the rain sensor make the decision when to water. The BlueSpray's way it not let such a dumb sensor make the decision. The controller has more brain than the sensor, hence the controller should make the decision. The sensor's job is just report its status.
Sounds like Hunter's manual tells you to connect the rain sensor between the valve wires and the common wire so that when it is tripped (open), it opens the circuit and the valves won't get energized.
With BlueSpray, you just connect those 2 wires to the rain sensor terminals and the controller will read it as on or off and act accordingly based on the configuration. The pump relay has nothing to do with it. If the sensor trips, the controller won't water hence won't turn on the pump any way.
I don't know what color the wires are, but you can use a mulitimeter to measure the resistance between those 2 wires. If the sensor is tripped (wet) and it reads 0, that means it's normally opened. Otherwise, it's normally closed. Connect those 2 wires to the rain sensor terminals.
Do not connect any sensor wire to the common wire terminal.