The actuator is a motor-driven lead screw or worm gear drive. It looks like there are only two power leads. There is a limit switch trick that requires two diodes that makes this work, and all of that probably is built in. Connect DC one way, motor runs until it hits one switch, then stops. It moves again only with the DC is reversed, when it runs until it hits the other limit switch. So, your control system needs to be able to reverse the DC polarity to the actuator to get it to move both ways.
If you want a push-push, auto reversing system, then you need a toggle flip flop function somewhere. If you want to wire up something, this can be a couple of transistors, or a single CMOS IC and some driver transistors. If you don't want to wire up something - what is wrong with you??? OK, seriously, you're in luck. An impulse relay does the toggle and power switching all in one part, no soldering.
If you want a more directed control system, with one lock button and one unlock button, that might be doable with the board in post #3.
*whap* ball's in your court.
ak