I am aware that there are many, many threads on this (e.g. this excellent thread). I may have missed it but I can't seem to find one that actually explains how to accurately report a frequentist confidence using the actual numbers contained in the interval.
So say I have coefficient from a regression $\beta = 3.4$ with $CI = [0.5, 5.6]$
Bayesian Credible Interval
If the CI were a Bayesian Credible interval reporting this interval is quite straightforward:
"Given the data and the assumptions of the model, there is a 95% probability that the true value of $\beta$ lies between 0.5 and 5.6"
Frequentist Confidence Interval
Now as we know when the interval is a confidence interval reporting it properly is more tricky. Based on what I've read I would hazard
"If we ran many experiments 95% of the 95% intervals constructed would contain the true value of $\beta$"
What confuses me is that the actual numbers in the CI do not appear in this interpretation.
How do I work either the mean or CI into my reporting of the effect while staying faithful to the frequentist view of probability?