Illustrating the burden of proof - pay me!

[The Architect Of Fate]

Here I think you're using haven't "met the burden of proof" to mean that a person's arguments aren't sufficient to persuade you

Or arent evidence period 

That seems perfectly sensible to me. After we've listened carefully to the other side, we all judge whether they've sufficiently made their case. Then we can decide whether it's worthwhile continuing the conversation.

Correct 

Some people, though, use a Burden of Proof Rule (perhaps handed down by Moses) to say that the side making the claim has to do all the work, and the side rejecting the claim doesn't have to give any reasons for their rejection. 

They do have to do all the work. They came to me with their rubbish, not the other way around and nor does the skeptic need to give a reason it's enough that the claimer has not proven their belief 

Again, I hold that thinking adults on both sides in a good faith conversation both have a case to make. Though I understand that the "thinking adults" thing may be a little too optimistic.

There is nothing bad faith about the above nor is it immature. You simply don't like it