I think most (sane) people implicitly parse the first sentence as:
anecdotal evidence of fairies.
Rather than:
anecdotal evidence of fairies.
I.e. the word anecdotal is used to express a measure of the quality or reliability of the 'evidence'.
Someone claiming they have seen a fairy is evidence of fairies, but it is also evidence for other (more likely) hypotheses which do not require the existence of fairies.
Your first statement seems to contradict your second.
But no matter, look at the problem another way. Is there evidence for fairies? Of course not. Yet by including all those possible conclusions for the person's anecdote in which
they concluded what they saw was fairies, you have implied there
is evidence, however weak and/or unconvincing, for fairies. But we know for all practical purposes, there isn't any evidence for fairies.
You don't have to use my, "evidence of a vision". I wasn't trying to argue semantics, I was pointing out a principle about what evidence is and is not. I can easily word it another way to make you happy. There is evidence that
an individual concludes is evidence of fairies, but it is not really evidence of fairies. You can word it anyway that you think meets perfection as you see it.
Except, you cannot say that because someone draws a conclusion about some evidence, that the conclusion becomes the evidence. And that is in essence what you are doing if you say that the person who believes what they experienced was X, means the experience was actually evidence for X.
Now to head off the obvious follow up problem here, at what point is something evidence for something, if we are all just collective individuals drawing conclusions? That is one of the things managed by the uncertainty language and principle of science. The evidence remains the same (excluding quantum observations). So all the wrong conclusions in the world don't make the evidence into something it is not.
For the vast majority of what we encounter, the evidence is either overwhelming, convincing, suggestive of, inconclusive and/or plausible, etc. So when I say there is evidence of X, I am still saying I see evidence and am drawing a conclusion about that evidence by saying it is evidence of X. If it is overwhelmingly likely the conclusion is correct, then we don't question the way the sentence is constructed. We agree that the evidence is evidence of X. This is true even if the evidence is only suggestive, or plausible but inconclusive.
But if you do not recognize that the way we construct the statement doesn't actually make something evidence for something it is not evidence of, you don't leave yourself that one particular thing you need to occasionally communicate, that there is
no evidence for fairies (and a few other things).
Most of the time, what you are saying is the most practical way of looking at the statement construction of a bad conclusion,
"but it is also evidence for other (more likely) hypotheses which do not require the existence of fairies". In that respect I totally concur. But regardless of how you word this,
you are still making a statement that includes a conclusion about the evidence for fairies.
If you remove that conclusion, and you just consider the evidence, is it really evidence of fairies? No. Did the evidence change because of what you concluded about it? No. Did it change because of what the individual who believed it was evidence of fairies concluded about it? No. Do we include a conclusion when we refer to evidence? Often, but not always. Is the conclusion actually part of the evidence? No. And that is where the problem here is originating. We are used to including the conclusion in the statement about evidence, but it is not mandatory to do so. The conclusion about the evidence is really something separate from the evidence.
You could put, "We think", "we believe", "we conclude" or something to that effect in the beginning of every statement that [Y] is evidence of [X]. That might solve the problem if people recognized that such an unspoken qualifier was actually in every statement that said [Y] is evidence of [X]. Then you have segregated the evidence from the conclusion and can see that it is separate.