Well, If it was me and I had nothing to lose.
I'd connect 12v and ground to it per the drawing. Then, with a variable resistor (> 1K) turned to the low side. Put the outer pins of the resistor on the 12v and ground, connect the centre sweeper to the signal wire.(again with the sweeper initially turned to the ground side). Then, slowly turn the resistor and see if anything moves.
Actually, after you connect the 12v, measure the signal wire and see if it has power. This will give you a clue as to whether the signal is grounded to move, or powered to move and will let you know where to start the sweeper at. If it is 0v, then start the sweeper on the ground side. If the signal is 12v, then start the sweeper on the 12v side.
Once you connect the resistor 12v and 0v, measure the voltage on the sweeper to confirm you have it turned the right way. What ever the starting voltage is on the actuator signal wire, is the voltage you should have on the resister sweeper pin before connecting it.
The main reason I would suggest starting with the resistor turned to the ground side is on the off chance the actuator uses a 0-5v signal. I doubt it since it is a 12v device, but better to be safe than sorry.
If the actuator does move, and while you are turning the resistor the actuator seems to reach full travel, don't turn the resistor more to "see what happens".
Maybe also look up the actuator and see if you can get any info on it. Might let you know more about how the signal is conditioned. This is just what I would do, so I take no responsibility if the smoke comes out....
