I'm having trouble understanding your example. What value is the 93.75% confidence interval for? Is it for the population mean? If so, why does the sample order influence the result?
In my example i(D) is a function of the data (a function of the ordering), and D is a random dataset. Since it’s iid by assumption (sneakily also assuming the probability of an exact repeat is zero), the probability that the interval contains all numbers is 93.75% (1-1/2^4). Otherwise it’s the empty set.
Unpacking that, suppose you have a real number m. The probability that i(D) will contain m (with D as the random variable) is 93.75%, so it is a valid confidence interval for m.
m could be the population mean, the population median, your dog’s age, whatever. The interval depends on the data, but not on the parameter, and the definition of a CI says that’s fine.
It’s a demonstration that definition of a CI alone isn’t really useful for reasoning about a parameter given an interval. You need to know more about the specific data generating process and function i that led to it in order to make sure it’s useful.
Or to put it another way, if i(D) is a function of the ordering, then isn't by definition the ultimate random process observed through i(D) not iid even if D is iid?
This paper describes the situation more thoroughly
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.691...
But, fair enough, I appreciate your sentiment about 'terseness'. One can only be so nitpicky with words when trying to communicate, before starting to sound like a criminal defense lawyer ...