The behavior you're seeing could be explained by mechanical issues with your Ball Count and Player steppers. The 2nd video seems to show that the Player unit doesn't advance reliably. You should be able to push any of the plungers all the way in, and release it and have the stepper advance cleanly to the next step or reset. If it doesn't do that reliably the issue is more likely mechanical than in the circuit that controls it. You could verify that with a bulb tester connected to the unreliable solenoid.

The smaller reset plunger you wonder about in the 2nd video doesn't pull out until the Player Unit steps out of the reset position.

The upper (Ball Count?) stepper shows signs of not being reliable in the past. Someone has stretched the springs to try to get it to over come some other mechanical issue. I'd replace any spring that's been stretched like that. Here is some info from the parts catalog:
Bally Stepper Full Reset 1976 catalog (resized).jpg
/Mark