Well it’s working but why. .
A theory: when you applied your voltmeter on the mV range, it would have offered a much lower resistance between the terminal you were testing and earth. That enabled a much larger current to flow, as demonstrated by the meter fuse blowing. But that current also flowed from source (battery) via all the connections up to your point of test. If there was a high resistance connection along that path, the excessive current could have burnt it down to a lower resistance that is now low enough for sufficient current to flow to light the fog light.
So, in essence, if this happens again, you need to focus on the wiring between this test point back to the battery, rather from test point towards the fog light.
Hope this makes some sense.
( For those interested in elektrickery: In my younger days working on the nation’s electricity supply network, underground cable faults were easiest to locate if they were either a dead short or completely open circuit, but anything in between was virtually impossible. One of the old tricks in changing the fault characteristic on high resistance faults was to lash a few electric fire elements to the end of the cable to draw plenty of current to either burn the fault down to a low resistance or blow it off completely. “A sh*t or bust method” Either way, it would then be a doddle to pinpoint using more sophisticated pulse echo equipment.)
