[EForEndeavour]

This is where I get lost:

> For a single confidence interval, you have either captured the mean in your confidence interval, or you've not -- there's no probability about it.

Isn't there? The underlying truth is that you either definitely have or have not captured the population mean in any specific confidence interval. But you can't know this truth. In the long run, if "a 95% confidence interval contains the true mean 95% of the time across an infinite number of replications of the experiment/study," then isn't it true that any single specific experiment's CI has a 95% probability of containing the true value?!

In my untrained mind, this is exactly equivalent to flipping an unfair coin with a 95% chance of heads. Sure, before flipping, the outcome of heads has a 95% probability. After flipping, you either get heads or tails. But if you flip a coin and hide the outcome without looking at it, doesn't it still have a 95% chance of being heads as far as the experimenter can tell?