Compressor units for a central air unit are quite simple: a compressor, a fan, the condenser coil, and under the access panel, a relay, and a run capacitor. The relay has the button, and when you turned on the thermostat, the relay activated, causing the button to be pulled in. That's normal. When the power or thermostat are off, the button will pop out. The thing you thought was a solenoid is the run capacitor. It is either a round or oval canister with electrical connections on one end. They come in single and dual versions. They come in single and dual units. The side of the can will be labeled with the capacitance value(s) in units of uF and a voltage. 370V is quite common. The classic symptom of a run cap failure is that the canister will buldge and not have a uniform shape anymore. Depending on the type, they cost $10 to $40.