The one on the right being held is the older style connector for the ignition switch....

Is that a fairly common type of connector and if so, does it have a name?

I tried for a long time to get one but never found anything, best of luck.

ok, new plug it is then.

These old wires are a pain to solder.

Use soldering flux.

The problem with using a soldering flux that you normally use for soldering a copper radiator is that as a liquid it wicks back up the multi strand wire and causes big problems for the future as it continues to eat away at the copper strands.
It does clean up the copper but I only use it if I can be sure of being able to get it all off before the soldering.
The flux cored resin solder will eventually clean it but heat gets to the plastic insulation before the copper is tinned.
Of course the multi strand wire is where you need it most because it also wicks the water which causes the black oxidation.  Hmmm that sounds a bit like a dog chasing it's tail. 

phosphoric acid cleans up copper but also eats lots of other things as well