Interpretaton of confidence interval

I have read two books that explicitly state that the $(1-\alpha )$% confidence interval should be interpreted as:

If you construct 100 such confidence intervals, $\alpha $ of them are expected to not contain the true population statistic and $(1-\alpha )$ of them are expected to contain the true population statistic.

and not as

There is a $(1-\alpha )$ probability that the true population statistic is contained in the confidence interval.

In my view, they both equivalent: If you make the first statement, you implicitly make the second statement. You are looking at any one arbitrary confidence interval, which in itself is a random variable, the generic confidence interval should, therefore, be subject to the second statement. Do things change when this random variable is actually realized?