The tests compute what a difference in magnitude of response such that, 95% of the time, if the measured effect difference is that large, the null hypothesis (that the responses of all subjects in both groups were drawn from the same distribution) is false.

I think that should be: The tests compute what is a difference in magnitude of response such that, if the null hypothesis is true, then 95% of the time the measured effect difference will not be that large.

Frequentist statistics cannot make the claim that with some probabilty the null hypothesis is true or false. Ever. You must have a prior and invoke Bayes theorem to do that.