As weef says, to confirm your diagnosis briefly dab a wire between a good earth on the chassis and the earth connection of the dim bulb - if it then lights properly it confirms a bad earth connection. Obviously, don't accidentally dab a grounded wire onto any other connection on the bulb: doing that would short your 12V supply to earth (and - hopefully - blow your fuse). So take care.
The earth wire on the dim side will run to a point where it connects to the chassis. Somewhere between (where you've measured 12V on the earth wire) and (the earth wire's chassis connection) there will be a bad connection. Most likely where the earth wire fixes to the chassis - either where it's crimped onto the eye or where the eye's screw goes into the chassis. But it could also be elsewhere along the wire.
The reason you're measuring 12V+ on the "earth" wire is that (when the bulb is switched on) 12V+ runs from the battery to one side of the bulb filament. If the bulb's earth is poor, you'll still be measuring a good proportion of that 12V+ even on the other (supposedly earthed) side of the same filament, where you should normally measure 0V.