The distribution of the population does not matter with regards to the meaning of the word "confidence". The statement "I’m 90% confident that the mean of the population is between 1 and 9" means that if all possible samples of size n (whatever size sample the statistician used) are chosen, and if a separate confidence interval is constructed for each of those samples, then about 90% of the confidence intervals will successfully contain the population mean (whatever it may be). The other 10% of the intervals will not contain the mean. Whether or not the particular interval (1 to 9) that the statistician found actually contains the population mean or not, well, all we can say is either it does or it doesn't!