Yikes. Well you have found the essence of the problem. Did you get my note, the FACTORY diagram for the ENGINE BAY has the bulkhead labeled wrong
The INSTRUMENT PANEL wiring diagram should be correct
So bouncing between the MyMopar diagrams:
http://www.mymopar.com/downloads/1972/72ValiantA.JPG
http://www.mymopar.com/downloads/1972/72ValiantB.JPG
and the 72 shop manual, and ignoring the lettering on the engine bay diagram bulkhead connector, we seem to have this:
START WIRE coming from start relay to bulkhead seems to be DARK GREEN/RED going to L in bulkhead. Changes to yellow at interior and goes straight to switch connector
BYBASS (IGN2) BROWN comes from coil side of ignition resistor and goes to Q of bulkhead, then goes to ignition switch connector BROWN
IGNITION RUN, (IGN1) DARK BLUE comes from switch side of coil resistor, goes to N of bulkhead and to ignition switch.
Does the harness "look" hacked, retaped or repaired? Could be miswired How long did you have the key to "start" when the brown got hot? How hot? "just warm"? or did it seem dangerous?
One warning---- NONE of the above circuits are FUSED. The only protection between the battery and ignition switch is the main fuse link out at the bulkhead, and if you short something, you can do a lot of damage and "meltage" before that link blows.
It would be wise at this point to unclip the sections of the bulkhead connector and inspect and repair them.
Read this MAD article for an overview of the what and why
http://www.madelectrical.com/electricaltech/amp-gauges.shtml