HI - You are truly lucky to not have the whole car burn up!!

The fire was from a dead short between the positive battery terminal and the block/ground. That provides about the same amp current needed to start the engine (500+) or to weld something.

The wire carries the charging current from the alternator to the battery, via the starter. Again, can be a lot of amps.

The wire from NAPA will work. Better a little fatter than too thin.

With the terminals on it, splicing should be unnecessary, a good thing.
I have used butt connectors on the battery-to-starter wire, they worked OK, but no splice at all is much better.

The thin red and thin gray (black) wires can be spliced up and routed again under the engine, OR - as recommended - re-routed from that gray connector on the firewall along the fender to the engine. Anchor them lightly on the fender well, and leave a good bit of slack to allow for engine movement.

If the black wire gets grounded, you get the "oil" light on all the time. If the red wire gets grounded, you get the "alt" light on all the time and the alternator won't charge.

Good Luck,

Bob

:>)