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"proof" means just that. EVIDENCE that can absolutely show that the question under consideration cannot have happened.
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"Proof" is indisputable evidence that something is true (or not, let's say for the sake of argument);
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Twice you define proof in terms of providing evidence, and yet you dismiss my providing evidence as a path leading to proof. We agree that proof is matter of sufficient evidence, but you dismiss my evidence examples for...why? Being too clear cut and obvious? I'm giving obvious, clear cut examples for just that reason.
In any court...or laboratory, for that matter...that I can imagine, proof of a MUTUALLY EXCLUSIVE alternative like I've given IS PROOF that the other thing didn't happen.
Indirect evidence leading to logical inference and deduction is simply HOW IT IS DONE when the thing itself is not directly observable, whether positive OR negative.
And the cases where this "You can't prove a negative" comes up is not in cases of two logicians arguing an abstraction. It's in matters of court cases, or sometimes in science related stuff, depending on genre, and so proof is never "absolute." It's "beyond a reasonable doubt" or "to a high degree of certainty."
What they, and you, apparently, are really saying when they say "you can't prove a negative" is either:
"You can't prove a negative...if you define proof as my arbitrarily chosen standard of proof, which will always be just a bit higher than whatever you achieve." (I think that's the "Real Scotsman" fallacy)
or
"You can't prove a negative...if you don't have any evidence." Well, duh. You can't prove a positive without any evidence, either.
This reminds me of another peeve of mine: "Absence of evidence is not evidence of absence." Actually, it often is. It may not be very strong evidence, and it's almost certainly not PROOF, but it often is evidence indeed. Again, this is typically how science is done when stuff isn't directly observable. If X happens, then we should be able to observe Y. We don't see Y. That doesn't PROVE X didn't happen. We may just have failed to detect Y. But it sure is evidence that X didn't happen, and we're going to continue experimenting to see if we find more evidence.