I don't know what you need...
Windshield wipers on a car use a DC motor. A DC motor (possibly geared-down or reduced with pulleys) and one or more position sensors might be appropriate for you too.
Stepper motors & servos are both "positional" motors. i.e. If you want to move a certain (variable/programmable) distance, they might be the solution. I don't think you need that... I think maybe you only need one or two end-points.
Servos are angular motors, and they can't rotate 360 degrees. Steppers can rotate more than 360 degrees, but they step 1.8 degrees at a time, so they are less precise (unless they are geared-down). A stepper also needs one or two home/position sensors so the software knows where it is when it starts counting steps. If a stepper motor looses a step, the software doesn't know 'till it gets back to the home position. A servo always knows where it is and where it's supposed to be and will self-correct.
Oh, servos have a driver circuit built-in so you only have to provide power and a position signal. DC & stepper motors require a driver circuit.