tom: would you briefly explain what, in your opinion, is the difference between a 'negative' and 'positive' claim? what does that distinction mean to you?
Negative claims are an absence, not a reworded positive claim. It is not "0". It is "-".
I can't make sense of this. What is an absence? What is a positive claim and how is it different?
Negative claims hold a special distinction. If I claim that the window is NOT open, it does not mean I am claiming that the window is closed. I am claiming that the window is NOT open.
You say that it holds a special distinction, but you're not saying what the distinction is.
How do I recognize a negative claim when I see one? Is it that it has the word 'not' in it?
What if I claim that the window is not not open? Is that a negative claim? What about 'it is not the case that the window is not open'? Is that a negative claim?
Also, what exactly do you think the word 'closed' means?

So, if someone introduces as a first claim, that there is no evidence that ghosts exist, it is their burden to prove that ghosts don't exist?
Yes. Every truth claim has a burden of proof. It sounds like you're saying that "Ghosts do not exist," and "I personally have encountered no evidence of ghosts," are logically equivalent statements. They are not.