I dont think you have anything to worry about.

Highly unlikely the ice maker has been damaged.

You didn't include the model number of the refrigerator but assuming it's a typical style of ice maker, getting cold enough is what makes them operate (run a harvest/dump and refill with water).

There's no water in it at the start. When the freezer section (and the ice maker) cool down sufficiently, the thermostat in the ice maker will trigger it to run a harvest cycle for the first time. Refill is at end of the cycle. When the water freezes, the ice maker's thermostat triggers another cycle for the first batch of ice and it then continues on automatically making batches of ice. There's a heating element in the ice mold to loosen the cubes for dumping. The scooper arm may stall briefly when it contacts the cubes, pressing on them until the mold heats enough to release them.

The first fill is typically short of a full level because the tubing that leads from the water valve at bottom of the refrigerator up to the ice maker is initially empty, so the first batch of cubes will be smaller than normal. Your instruction manual likely advises to throw out the first few batches of ice until the supply line has been flushed clean after several fills.

If the supply water is turned off but the ice maker is turned on, it will cycle each time the mold gets cold enough to trigger the thermostat ... which is essentially no different than normal operation except there's no water freezing in the mold.