I didn't read every word so if this has been covered forgive me. Lets clarify something here, "Brazing" is melting a brass rod to "glue" two pieces of metal together. I say "glue" because that is literally what it is doing, it's simply bonding them with brass which melts at a much lower temp than the metal. The metal is NOT melted together like welding. Where as welding is literally melting them together and making them one piece, brazing is not.
Brazing and welding are two completely different things, but use the same tool, a "torch", it is used for both brazing and welding with a metal rod (or no rod at all if you have a tight enough fit). It uses oxygen and acetylene gas to create a fire at the tip that is hot enough to melt the brass or metal.
You could braze with a TIG I assume, I have never used one but I assume. You can also braze with a MIG. We have one a work, a MIG (which of course it isn't a "MIG" anymore but that is a whole different thread) converted to braze. It's pretty cool, we use it on cars that had stock joints that were brazed as some late model cars still use it here and there.
Here is a little video on the subject. I am not saying everything is correct, I haven't watched the whole video but just to get an idea of what the differences are.
Braze vs. Weld - YouTube
And yes, mufflers were gas welded on for years. In fact I had a shop next to a muffler shop and saw it every day for years.

I am gas welding again a little and dig the trip back in time. And yes it was done years ago with coat hanger, I welded a lot of stuff using coat hanger.
Brian