I'm betting the water is coming into the attic on the outside of the furnace exhaust... not leaking down inside the duct.

That was definitely what was happening with the first leak. And maybe you're right, and it's happening again. Just seemed odd that the outside of the duct was dry everywhere except that seam.

So if it is leaking again on the outside, despite the previous fix, I'm going to have to contact another roofer to repair this properly. What should I ask for/expect to be done to ensure this is done properly?

I wonder if what I'm seeing is actually condensation inside the pipe that's leaking out.