You could use a multimeter tied to ground to see if voltage is going to the door instead of touching it.

Voltage test. But you already know voltage is going to the door. What you don't know is where it's coming from or how much voltage. Test it to determine how much voltage and what kind.

When you fix the problem, properly ground your game so stray voltage goes to ground like it should, rather than into the person touching the game.

It's most likely 115V from the start relay and it's shorting at the credit button or on the coin switches. They need to be replaced/rebuilt/adjusted. There are a few other possibilities, but this is the most likely reason based on OP's post. Moving it slightly misadjusted the insulation

This post in one of vid1900 guides has information on properly grounding your game: https://pinside.com/pinball/forum/topic/replacing-line-cords-plugs-wall-sockets-vids-guide#post-1945124

Start with one of these.

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I agree with the safety aspect of having voltage go elsewhere, but is it that common to just ignore that something is going to ground that probably shouldn't be?

The problem should be fixed, not ignored. Properly grounding the game will not fix Mahoyvan 's problem. I did not mean to imply that it would. It will prevent someone getting a shock when touching the game.

Same issue with the GFCI outlet tester suggested above. That does nothing to solve the problem of voltage going somewhere it should not be going. It renders that stray voltage mostly harmless when the game and outlet (and house) are all properly grounded.

Don't touch the door, don't touch the door, DON'T TOUCH THE DOOR...

Something I recommend for safety is a Ground Fault outlet, breaker, or power strip. That outlet most bathrooms have with a Test button detects if power is leaving the normal system and routing through your body, and turns itself off.

https://en.wikipedia.org/wiki/Residual-current_device

I usually cut the 110v wires from the front door tilt switch, reconnect them well inside the cabinet, and shrink wrap them so no shock is possible. Stupid to have 110 so close to a metal door just for a tilt switch in a home environment!