Lack of evidence is always weak evidence for evidence of lack, and pretty strong evidence if you've looked hard enough. If you've trawled through loch Ness with a fine toothed comb for decades and haven't seen Nessie, well then that's pretty good evidence for lack of monster.
You're confusing correlation with causation. Regarding your point about searching - Have you ever lost your keys, searched for a long time without finding them, then found them by happenstance at a later point when not looking? Lack of evidence is not evidence of lack.
I am saying your definition of evidence is overly strict and at odds with both how we use the term in common speech and with what is useful. My definition of evidence for A is an observation B such that P(A|B)>P(A), and with this definition, lack of evidence most certainly is evidence of lack.
Yes, I understand what you mean. You are confusing your particular experiences with the common speech of everyone, which is a logical fallacy.
You can read about the cultural and traditional idiom I wrote at the wikipedia page for "Evidence of Absence" where the first paragraph mentions, "Per the traditional aphorism, 'Absence of evidence is not evidence of absence,' positive evidence of this kind is distinct from a lack of evidence or ignorance[1] of that which should have been found already, had it existed."
There is further information in the wikipedia page for "Argument from ignorance" that shows why your use of evidence is also a logical fallacy. You can infer from indirect evidence, but that doesn't prove a fact.
While indirect evidence may lead one to to believe a fact has been proven, that is not what happens. You can read some of the legal ramifications of using indirect, inferential, or circumstantial evidence to convict beyond a reasonable doubt at https://www.legalzoom.com/articles/why-cant-some-juries-conv....
