It is distinctly different, but I have to think really hard to follow the practical difference between "If we ran 100 more identical experiments, 95 of them would estimate a population mean result somewhere in this range", and "There's a 95% probability that the population mean falls somewhere in this range."

The thing that makes it hurt is that what seems like a minor rewording of that statement, "We're 95% confident that the range we calculated contains the true population mean," would be correct. (It's not the definition of a CI, but it is implied by the definition of a CI.) And that the reason why one is true and the other is false is down to the technical distinction that the true population mean is a discrete value and not a draw from a random variable, so, from a strictly mathematical sense, it is nonsensical to apply probabilities to it. By a the same token, an even more subtle rewording gets us back to falsehood: "There's a 95% chance that the range we calculated contains the true population mean."

Which, delving into to that level of hair-splitting makes for interesting math, but also leaves me with the opinion that, sure, the most common intuitive interpretation is wrong, but it's wrong in a way that isn't really of much practical importance.

By contrast, the most common intuitive interpretation of the p-value is disastrously wrong.