Would an electromagnetic clutch such as this one count as a solenoid for power limitations?
You’ll have to ask on the Q&A to get an official ruling if you want to use it.
That said, in the past FIRST has ruled that any sort of electromagnet that creates motion is considered an electronic solenoid. So, based on that history, I would say it probably would be considered an electronic solenoid.
The actuators on the page linked are going to be “interesting” to get past inspection. You’d have to show that they’re legal–a 24V 7W clutch is going to raise a few eyebrows.