If it can be, how can the "Absence of Evidence" be "Evidence of Absence?".

You've said this before, but it's wrong. You can have evidence for something that's wrong. That just means the evidence wasn't sufficient to rule out all other possibilities, it was interpreted incorrectly, or something else.To go back to the snowy camper analogy. Say you wake up and find tracks by your tent. That is evidence that something walked by your tent. Since the tracks are not preserved enough to show you exactly what walked by, it is equally evidence that a person walked by as that a bear walked by. You would then need other evidence to narrow down the options.If there is a lack of evidence, that is a very good reason to tentatively reject the claim. Once some evidence one way or the other comes along, you can adjust your belief.Belief is not absolute, it can change.

You're right, which is why your statement that saying evidence that there is no water on the moon means we can't ever find other evidence that says there is water is wrong. We can never prove a lack of something (except by proving a mutually exclusive alternative), all we can do is amass evidecne and weigh it. If there is no evidence, the scale tips, ever so slightly, to nonexistence. Well, for most things, there is no complete lack of evidence. There si something on whcih to buold a tentative conclusion. For the moon water thing, we found hydrogen, which is slight evidence for, we also know that in a vaccuum, water quickly boils away, we also knew there were permanently shadowed craters and that if ice got buried beneath regolith, it might be protected slightly, and that water can react with other compounds to create a new compound that has "water" locked inside. All of that was evidence that was weighed, along with every new study of the moon. So, it wasn't a lack of evidence, by any means, it was a pile of evidence, much of which was ambiguous, just like the tracks in the snow. In an absolute sense, a lack of evidence can't be evidence of anything...however, it is compelling reason to reject belief in the claim, and can be held up as a rational reason for dismissing the claim...in colloquial language, it's evidence to be weighed and is exactly what you would expect to find if the claim were, in fact, false.

No, it's not proof, and can't give you 100% certainty...however, it is evidence to be weighed. If your hypothesis is "There is no pen on the desk," one of the predictions of that hypothesis would be not finding evidence of a pen on your desk.The experiment (looking at the desk) then supports that hypothesis. It's confirmation, but not absolute proof.

I guess the part of "on the desk" sort of implied looking at the desk.So then, to be more exact, the prediction would be: "If I look at the desk, I will see no evidence of a pen."