I probably never would have even thought about it, if it wasn't for the name!
The material that a rope is made out of, doesn't really matter. What matters is it being built up of smaller fibers.
But for many European languages, the word "wind turbine" would be translated to "windmill", even going as far as labelling parts with the word. Never worked with wind turbines, but been curious about this for years since the language is so systematic. A documentary I recently watched had a Danish wind turbine marked as "Mølle 1" or something like that, translating to "Mill 1".
I'd be surprised if Norwegians didn't use the word "kabel" (cable) to refer to the thing that carries ski lifts.
The truly amazing thing, though, is the safety record: they have a mechanism to switch from a fast rope to a separate, slow rope and back. This transition (and, indeed, the grip on the rope) manages to have an excellent safety record despite being relatively complicated and dealing with snow and ice.
the grip itself is spring loaded and the spring force is measured every time the grip passes through a terminal. if a grip is weak, the lift stops.
if you look closely at a detachable grip, you'll see two wheels. those go into a funnel that squeeze open the spring loaded grip to release the haul rope. the chair or gondola cabin then is guided with one of those wheels while powered by the 20 or so small rubber wheels mentioned above that move the chair through the terminal. reattachment works the same, the wheels progressively speed up to the haul rope speed (5-6 m/s) as the funnel opens releasing the grip to snap back onto the haul rope.
position sensors throughout the terminal and at the towers stop the lift if anything is out of position. (or a grip is weak, as mentioned above)
and like any safety critical reliable system, a lot of hardworking humans check the system regularly.
In the station it detaches from the cable and just rides on the flat track, and then motor-driven tires push on the gondola to propel it at a controlled speed through the station.