"The probability of this parameter being in the interval" is what's called a credible interval, and it will coincide with the confidence interval if you assume a uniform prior, that is, if before running the experiment you figure every result is equally likely.
In general, frequentists just don't like to talk about the properties of this particular sample, only about long-term frequencies – hence the name. Why? Because they object to the idea of probability as a degree of belief rather than as an objective measure, and given that attitude the statement that "there's a 95% probability the parameter is in this interval" doesn't make any sense: either it's in the interval or it isn't.
> it will coincide with the confidence interval if you assume a uniform prior, that is, if before running the experiment you figure every result is equally likely
Huh? Unless you can bound the set of potential results, this isn't possible. Say I want to estimate the half-life of some material (bounded below, but not above). A uniform prior doesn't exist. How will the credible interval relate to the confidence interval?
In general, frequentists just don't like to talk about the properties of this particular sample, only about long-term frequencies – hence the name. Why? Because they object to the idea of probability as a degree of belief rather than as an objective measure, and given that attitude the statement that "there's a 95% probability the parameter is in this interval" doesn't make any sense: either it's in the interval or it isn't.