Converting an AC switch signal into a reliable Digital ON/OFF signal
Good question. Here's what I would do: First I would build some kind of AC peak detector, and maybe even give it a threshold voltage so that it doesn't detect a "peak" unless it's >15v or something.

Then I would basically try to model the score reel using software. It will probably not be perfect, but it should be able to get pretty close. I would assume that the wheel needs to be energized for X peaks before it "latches". I would also assume that there's a certain reset rate, where if there are no peaks then the solenoid retracts. This assumes the wheel is spring loaded to return to an integer position if it gets a partial trigger. To model that, I would decrement the peak counter at a specific rate over time.

Once you register a trigger, just wait for the peak count to go back down to zero and then reset the whole thing.

Another option would be to attach some kind of encoder to the first disk of the score reel. I think that's more likely to get a perfect count over a long period of time.