That's really strange that it happens in 3rd and neutral but not 2nd. Seems that would rule out the rear axle/diff.
I'm no drive-line expert Dave but isn't the first pic correct too? Thought the problem was if the motor was flat and the rear axle was tilted or vice versa -- that was the no-no. Reason I thought the first pic is correct was that I remembered the old model A's my Dad used to have where the drive line was flat back to front ?? (but maybe those are a whole different ball game - ah - maybe those oldies didn't have U joints).
Know anything about the C6? Come out of a wreck? (thinking maybe something bent inside or a bad bearing if it was wrecked while in 3rd gear)?? but that doesn't seem likely since it happens in neutral.....very strange.
Just guessing really - hopefully someone will know a simple fix.
Keep us posted - interested to see what the problem turns out to be.