This is very similar to Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?
Your first property is more strict because it requires that the confidence interval contains 95% of the time the true parameter, conditional on the true parameter. The second property does not specify this conditional probability.
(The first property is not so explicit but you have this contrast between ”the same experiment is repeated a large number of times” and "After a large number of independent situations".)
This graph below comparing a confidence interval and a credible interval might be helpful to show the difference. In the left image we see that a credible interval will not be containing the true parameter exactly 95% of the time conditional on the particular true parameter value. For some parameters it will be more for others it will be less. But on average (averaging over the prior distribution of the parameter) it contains the parameter 95% of the time.
