I always use only the game switch and never have a problem, it sounds to me like you just experienced the games power supply storing a charge in the solenoids power section capacitors. No worries, just allow the game to self discharge for a little while before working on it if you are concerned, and stop throwing your tools around

Do you have the correct power switch in the game.It should be spdt to disconnect both leads. Sometimes they get replaced with one that only breaks one line.

I always physically unplug games I'm working on. I don't want a shock, and my game doesn't want a short.

Unfortunately that won't solve the issue of charged capacitors, which--depending on the circuit--can take a while to discharge. The safest route is to either get a capacitor discharge probe or you can put your meter on the cap and discharge it that way. The latter technique requires a high-resistance meter and is applicable only to low voltage caps (in this instance solenoid voltage would be considered as low). Do not attempt this with a cheap meter.

Does turning the switch on, with the game unplugged do anything to discharge caps? I suppose that would also depend on the circuit.

As a general rule when dealing with machines that have AC to DC power supplies, either wait a few minutes with the power off before servicing, or short the capacitors with a small screwdriver (not possible in many cases.)

THIS DOES NOT APPLY TO HIGH VOLTAGE (CRT's, DMD's). There are proper tools for working on those.