No, I'd say you have a positive wire to one of the lights that is "grounded", which gives you a dead short. Probably, the insulation is old and worn and a bare wire is touching the frame somewhere.
Now, how to find this. If you have a continuity meter or even an ohmmeter, remove all the lights. Connect the meter to the positive and negative leads to the running lights, if you have continuity (or very little resistance) a positive feed is grounded somewhere. I haven't looked at this wiring, but if you can get to individual lamp inputs, check each of them for continuity with ground. If you find one, that's the problem.
Other than that, just carefully trace the positive feed(s) from the plug back to each light to see if you can find the "fault". I think it will be visible. I'd actually look first behind each light - I think the fixtures are pretty easy to remove and that gives you access to each positive lamp feed.
- Jack