Anyone Know What this Connector is Called?

Mattax said
OK. The wobble was why it got hot. If you want to disect it further and post photos, maybe it will help others.
When you install make sure the studs are insulated from any metal dash parts. Some use fiber washers, others go through the circuit board. I don't know your cars dash just mentioning it.

The bulkhead connector - yes. If you can reinstall the terminal so it doesn't move, then maybe its fine. If it can move then the vibrations could cause the screw and crimps to lose grip. A lot of us are dismissive of the wire supports, removing them when they get in the way of repairs and maintenance. Only years later we find out the hard way that the guys who built the car had a reason for putting them there. I for one am a lot more humble these days.

The bench test was a way to reveal resistance if you were reusing the old one.
The concept can be used broadly so its worth reviewing anyway.
Remember that current is electrons moving.
Voltage is level of energy.
When flowing electrons encounter resistance, power gets used and the energy level drops.

On a slightly different subject-
I don't know how the cables are laid out on a '63.
It may be worth looking into using a later style positive cable that has a second wire going directly to the starter relay.
The only thing it might do is provide a more direct connection in a cooler environment. On a slant six this is pretty open and maybe doesn't make a difference.
The other thing to consider is installing a fusible link. The purpose of the link is like a fuse. Its a smaller gage wire with special insulation. The idea is that it can handle occassional, short periods of excessive current ( 40 amps) but if the battery is accidently grounded, it will melt immediately.
It probably would not have helped prevent the overheating failures you saw.
I can't picture this 'webbed' wire you inherited and removed but have to think whatever caused that damage also may have damaged the ammeter stud connection.

The easiest type of link to add will be one with a ring terminal large enough to go on the relay stud.
In the Chrysler Tech booklet we can see a double link used in '65 for the Fury that also protects the horn relay.
Electrical Accessories & Service (Session 207) from the Master Technician's Service Conference

I don't know if anyone repops that or the Valiant/dart version, but if you really like the idea you can buy a 16 gage link and make your own.
Here's two versions with a ring terminal
View attachment 1715691004

May be a silly question but for the wire supports you're referring to the sort of hooks that hold the wire harness in the idea path under the dash correct?

The second cable going directly to the starter relay, I believe mine has that, when I originally replaced this positive wire months back before this overheating happened there was a much smaller black wire crimped onto the positive wire behind the ammeter. If I remember correctly it was connected to the starter relay and I did put it back together in the same fashion following a wire harness diagram for a 1963 Dart off of Allpar.com and was going to obviously do that again. I'll see if I can pull up the diagram when I get home from work and post it here as it may be helpful for y'all to see what I'm working with.

Another thing comes to mind and I'll highlight it in the diagram later but the point where 5 wires converge got damaged and had to be twisted and electrical tape and heat shrinked back together. Could this be a major concern going forward? My neighbor was the one to do this and I had no clue at the time that that might be something worth replacing the whole dash harness over.

The "webbed" wire is indeed the same wire I"m working with now that fried so knowing what info now I wouldn't be at all surprised if it's the same problem showing its face again. By webbed I mean it clearly had overheated and the red casing over the wire split open and looked like a spiderweb showing exposed wires from when I purchased the car.