Anyone Know What this Connector is Called?

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?
yes any sort of support. Wiring should have some support and often also 'stress relief loops'. Stress relief is just a little slack in the wire right at the connection. You'll see onthe valve cover and probably on the alternator iteself there are supports for the wire.

The second cable going directly to the starter relay, I believe mine has that,
OK. I just didn't know what '63 had.

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.
smaller black wire crimped to the R terminal.
Not sure. Horn, clock, and later on I think I've seen convertible and hazzard wires connected on the battery side of an ammeter.

Show us what you got from Allpar.
The best wiring diagrams are the ones in the Dodge and Plymouth Service manual.

Positive wire implies the other wire is negative. The other wire is also positive. Its just the connection to the main splice. That whole line is positive.
Nothing in that circuit is a grounding wire. In fact touch any of those wires to ground and the battery will try to fully discharge - picture an arc welding situation. That's why later on they decided to put a fusible in the connection from battery.


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?
oh YES!!!
You don't have to do all the fancy and clever things Bill Grissom described but you absolutely need to make sure that splice is clean and all the wires very well connected like he said. That's the main splice where the alternator output and the battery output join the main circuit feeds. (see the diagram posted earlier). Cut the heat shrink off. Clean it. And if anything is loose or permanently damaged, at minimum solder the splices. I'd actually recommend using a crimp splice and soldering. Like shown here Inline wiring splice clips........from the Dodge RAM service manual