I have a sinking feeling that I’ve lost my grip on whatever was the point of the thread! But let me try to say what I think is the case without, I hope, making matters more complicated than they already are, and thus providing you with another, well-justified, occasion for gently poking some fun at my effort!
If I understand you correctly, you make two claims:
absence of evidence is always evidence of absence; and
absence of evidence is sometimes proof of absence.
You then go on to say that when people say, “absence of evidence is not evidence of absence” (i.e. when they deny no. 1) what they are really doing is to assert “absence of evidence is not proof of absence” ( i.e. they deny no. 2).
Having kicked the ideas around in my head for a few minutes, I am inclined to think that you have a point. If there is no evidence for the existence of phenomenon or event X, then that, just by itself, constitutes some evidence for the non-existence of X, though not necessarily conclusive evidence. (Perhaps we haven’t looked hard enough, or we’ve looked in the wrong places.) However, in certain situations (e.g. the question is whether or not there’s an elephant in the room, and the room is sufficiently small and we’re not talking about some species of microscopic elephant!), then the absence of evidence for the presence of an elephant (e.g. you can’t see it; there’s nowhere for it to hide, etc. etc.) is not only some evidence of the elephant’s absence but conclusive evidence; in your words, proof.
Thank you for clarifying matters.
With every good wish,