I'm sympathetic to both views.

I have encountered a number of disputes that revolve around using these two different senses of the word, and am nonetheless blindsided by them consistently.

I try to always specify the strength of evidence in some sense when using the word. I think when I do use it unmodified I tend to use it in the technical sense (including even weak evidence).

It would be odd if 'evidence' excluded weak evidence, since then 'weak evidence' would be a contradiction in terms, or you could see people arguing things like "When I said 'weak evidence' I didn't mean the 1e-15 bit case, since that's not evidence at all!"

Hmm. Maybe the strength of the evidence isn't the right thing to use, but rather the confidence with which we know the sign of the correlation.

I would if I were talking to a Bayesian, interpret it as meaning something where a "B is evidence for A" if rough calculation shows that P(A|B) > P(A). I don't generally expect rationalists to even mention individual data points unless P(A|B)/P(A) is large, but if someone else gave the data as an example, then I wouldn't expect it to be necessarily large if a Bayesian referred to as evidence. So for example, I could see a Bayesian asserting that the writing of the Bible is evidence for a global flood some 5000 years ago, but I'd be deeply surprised if a Bayesian brought this up in almost any context because the evidence is so weak (in this case P(A|B)>P(A) but P(A|B)/P(A) is very close to 1).