Yea, you really need the:
Force required
The motion required: e.g. straight line
How much motion is required: e.g. length of travel
And what sort of motion is needed. e.g. push and release
How fast?
There is also the idea of just simple rotary motion and sensors.
You also need to look at the speed at which this is to happen.
The "fish scale" or "luggage scale" or "spring scale" is a way of measuring force. Another way to estimate is with weights. e.g. to measure the force required to pull a drapary closed, attach cans of various weights until it moves. So, you can get the force in lbs or ounces.
It you wanted a motor to move the cord, you would need torque and those units are in oz-in, for instance.
A "door lock actuator"
https://auto.howstuffworks.com/power-door-lock3.htm might be a place to start. This
https://www.amazon.com/High-Power-Door-Lock-Actuator/dp/B0002KR9PW has on 3/4" of travel.
An AC synchronous motor and a rack and pinion with limits might also work.
Adjustable clutches like these
https://www.sdp-si.com/eStore/Catalog (S98CA6-MOC200424 ) protect what your driving
There are so many different ways.
Here **broken link removed** is a little gizmo. You mount the switches to limit the travel with the proper controller.
There were two systems I was involved in. One used a 24 position rotary solenoid, That was a strange beast.
In another case, I had to rotate four 2"x2" filters into an optical path. I placed a roller micro switch on the edge of the wheel and an indentation for each filter. I used 2 "hall effect" sensors to get the position "00" "01", "10" and "11". My controller knew when the motor was moving or stopped and I could tell the motor to ignore the switch.
Other systems were shutters.
Your motion could be press and release. much like the door lock.
Slide projector mechanisms were very slow and were prone to jamming. The one I had actually moved the slide and then put it back into the carousel,
Stepper motors are another option.
I'll throw air as an option too. e.g. a double acting air cylinder. Tricky to set up.