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In my mind, "standard deviation of the mean" is a perfectly fine, unmixed term.
But you seem not to agree. See below:...
ok, so who, among "NIST, CDC, professional statisticians, etc." uses it?
From my own experience, "standard error" is indeed used more commonly when talking about the standard deviation when applied to a value (like the mean) calculated from a sample.
As already pointed out, the central limit theorem does not apply to all
distributions. You must state the theorem carefully. In the case of the
chi-square distribution it is easy to demonstrate by example with MathCad
or a similar program that the large N limit is not Gaussian.
Would it be OK to say that the CLT applies to any distribution of x
that is confined to a region between x1 and x2? By this I mean that the
probability density function is zero a all t x<x1 and at all x>x2. That
would cover nearly any practical distribution I can think off.