In the first case, you have independent evidence that the conclusion is false

"Independent evidence" is a tricky concept. Since we are talking Bayesianism here, at the moment you're rejecting the argument it's not evidence any more, it's part of your prior. Maybe there was evidence in the past that you've updated on, but when you refuse to accept the argument, you're refusing to accept it solely on the basis of your prior.

In the second case, you're saying "I have already concluded that your conclusion is false because I have concluded that mine is true."

Which is pretty much equivalent to saying "I have seen evidence that your conclusion is false, so I already updated that it is false and my position is true and that's why I reject your argument".

I see that there are two ways to interpret it.

I think both apply.