Mike,
Sorry to hear about your wire.  Good that you can move fast!

Using the Service Manual (SM) diagrams, I've compared my 3B harness to the 3A.  At the starter it seems very similar.  The only real function I see to that wire that goes to the Ammeter is so it can register Amp draw (negative when you hit the starter rod and positive when the engine is revved). 

On both, that terminal on the back of the gauge shows just 2 things connected:  The + wire (after passing the starter) and the horn (it needs to be hot always vs. switched). 

The other terminal should have a wire that runs to your ignition and the other wire to the regulator.  All wires need to be insulated from the clip that holds the gauge up against the "dash."

It may be possible there was a grounding on that wire.  It may be possible the starter pulled too many amps and overloaded the wire.

Before you disconnect, using an Ohm meter, check to see if you have continuity from the end of the burnt wire to the metal body, i.e. the bracket holding the gauge or the back of the dash somewhere on bare metal.

You may want to test the starter (I'd take it to a shop so they can put it on a machine).

You're correct on the flow from the ammeter supplying the rest of the vehicle with juice.  The ammeter must be in in line so you can see drain/charge cycles.  The light switch has a breaker so if there were a short, all circuits after that would be isolated when the breaker opens.  The ignition circuit isn't connected to that breaker.  It goes from the ammeter to switch and then directly to the coil etc.

I found out that some vendors order harnesses from Vintage Wiring and then sell under their name.  I personally went with a Vintage Wiring harness and their support is unrivaled.  I got lots of help. 

The difference in wires between voltages is a 6V typically has larger wires to avoid restricting flow (not sure about all wires but the battery cables are bigger).  12V wires can be smaller.  I didn't know an M38 was 12V (I would have assumed 24V) but from this perspective seems irrelevant; you aren't doing a military vehicle restoration.

I'm only guessing but a 12 or 10 gauge wire should work.  Before hooking it up, you may still want to figure out why it burnt - to avoid repeats.

Good.  Sounds like you know where it began.  A wire replacement should fix you up!