Research!

As a friend told me - I really should have done a little research before soldering away...

Anyhow;
The problem is easily and cheaply fixable.
And it will be fixed properly and according to how pinball machines are supposed to operate.This picture blatantly stolen from Marco Pinball pretty much explains everything needed to know.

1) Lamp socket with diode attached. I'm frankly a bit ashamed I didn't research the purpose of the extra lug when I checked the connections....

What I did was soldering my ground cables directly to the outer pin, efficiently bypassing the design of the lamp holder. In other words, I did exactly what this image told me not to.

2) The problem explained in detail. On my machine the row and columns are "swapped",
meaning power is coming from the rows and columns are ground. 

The solution is pretty easy to put in place, so now I'm off to shop for a bunch of 1N4004 diodes - which, luckily, are very cheap compared to the the cost of converting to all-LED...