Extraordinary Claims Require Extraordinary Evidence

[robvalue]

The phrase is only an informal rule of thumb, I think you're maybe taking it a bit too seriously.

Extraordinary, as I would use it, means highly unusual.

If the claim is something highly unusual, then anything other than highly unusual evidence is not likely to suffice. If the evidence for the kind of claim was in fact more commonplace, then the claim itself would no longer be so unusual.

If you're talking about science, you don't need this phrase at all. Any claim must make specific testable and falsifiable predictions. This is where anything to do with religion falls down instantly because it makes no such things.

For history, all you can do is examine the available evidence to decide what the most probable sequence of events were. Accounts that mention things that are not known to be possible carry an extremely high probability of being inaccurate. In fact, talking about such things is an indication that the person is most likely deluded or on the make, in my opinion. All we have to compare to is what we know about reality, and the further the accounts deviate from this, the more likely it is that there are simpler explanations than "something totally unprecedented happened". There is far more room for interpretation with history, but any opinions not supported by relevant evidence are not worth much.

But regarding history, there is no way to reach firm conclusions like there is with a scientific hypothesis. Historians can evaluate the age of evidence, the style used, the language and so on, and then offer their best estimate as to what happened. With science, there is no estimation or room for opinion, there is only what can be tested and demonstrated. (Of course, people can have different ideas about as yet unproved hypotheses.)