So far I got the point; it seems the Anderson–Darling test and the Cramér–von Mises criterion are similar, just based on a different weighting function $w$. Also there's a variant of the Cramér–von Mises criterion named the Watson test.
Basically I have two questions here
There are not many Google results about these two methods; are they still state-of-the-art? or replaced by some better approaches already?
It's a bit of a surprise, as according to this paper on power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson-Darling tests, AD is performing quite well; always better than Lilliefors and KS, and very close to the SW test, which is specifically designed for the normal distribution.
What is the confidence interval for such tests?
For the AD, CM and Watson tests, I saw the test statistics variable defined on the wiki pages, but didn't find the confidence interval.
Things are just more straightforward for the KS test: on the wiki page, the confidence interval is defined by $K_\alpha$, which is defined from the cumulative distribution function of $K$.