I thought that "evidence", unmodified, would mean non-trivial evidence; otherwise, everything has to count as evidence because it will have some connection to the hypothesis, however weak. To specify a kind of evidence that includes the 1e-15 bit case, I think you would need to say "weak evidence" or "very weak evidence".

But I'm not the authority on this: How do others here interpret the term "evidence" (in the Bayesian sense) when it's unmodified?