I know roughly and informally what a confidence interval is. However, I can't seem to wrap my head around one rather important detail: According to Wikipedia:
A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained.
I've also seen similar points made in several places on this site. A more correct definition, also from Wikipedia, is:
if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level
Again, I've seen similar points made in several places on this site. I don't get it. If, under repeated experiments, the fraction of computed confidence intervals that contain the true parameter $\theta$ is $(1 - \alpha)$, then how can the probability that $\theta$ is in the confidence interval computed for the actual experiment be anything other than $(1 - \alpha)$? I'm looking for the following in an answer:
Clarification of the distinction between the incorrect and correct definitions above.
A formal, precise definition of a confidence interval that clearly shows why the first definition is wrong.
A concrete example of a case where the first definition is spectacularly wrong, even if the underlying model is correct.