Possible misunderstandings of the confidence interval definition keep on bugging me from time to time, and I want to get this sorted.

More specifically, I refer to a misunderstanding I believe well explained on the [mobile-Wiki page](https://en.m.wikipedia.org/wiki/Confidence_interval), section "Misunderstandings". (also in the [regular version](https://en.wikipedia.org/wiki/Confidence_interval#Meaning_and_interpretation)) 

For clarifying the correct meaning of the concept, the following paragraph ensues: 

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> A $95\%$ confidence level does **not** mean that for a given realized interval there is a $95\%$ probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).[10] According to the strict frequentist interpretation, once an interval is calculated, this interval either covers the parameter value or it does not.
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So, if I repeated the experiment and calculated appropriate confidence intervals $n$ times, I would capture the parameter of interest 95% in the limit.
But picking one realization, I cannot make a probability statement.

It seems analogous to me of saying:

*We have a machine that, when producing $n$ balls, tends to produce 95% black balls as $n$ grows. Yet, if somebody asks you, as the machine is about to produce a ball, or after a ball is produced and kept hidden from you, what colour will it be, you cannot make a probability statement.*

I find this very puzzling.

Alternatively, I find this additional source of confusion:on the same page I read the following statement:

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> The confidence interval can be expressed in terms of a single sample: "There is a $90\%$ probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter
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I fail to appreciate the subtlety: if a statement is made on the confidence interval, saying it will encompass the true value of the population parameter, is it not necessarily also a statement on the latter made?
I mean, if an interval contains a real number $a$, is this not equivalent to making a statement about the number, saying the number $a$ lies in the interval?

Thanks a lot. I checked other answers on the point but failed to understand.