> Yes. The reason the various sigma values have the numerical values they have is because they represent integrals under the normal distribution, either one-tailed or two-tailed.

No. I defined sigma, that is, standard deviation, fully precisely and correctly.
The normal distribution has nothing to do with that
definition. And the standard deviation is a number,
just a number, just as I defined it as

σ = E[(X - E[X])^2]^(1/2)

which clearly is just a number and not an area.

Or, for random variable X
with cumulative distribution F_X, that is,
for real number x,

P(X <= x) = F_X(x)

we have, with notation from D. Knuth's TeX, that

σ^2 = \int (x - E[X])^2 dF_X(x)

This integral need not be in the sense of
Riemann (i.e., freshman calculus) because
dF_X is a measure on the real line;
so, the integral is in the sense of
measure theory (see any of Rudin, Real and Complex
Analysis; Royden, Real Analysis;
Halmos, Measure Theory; Loève, Probability
Theory).

Sigma is defined for any random variable X or its
distribution provided that E[X] exists and is
finite. Again, a "normal" or Gaussian assumption
is not necessary. So, sigma is defined for
discrete distributions, the uniform distribution,
the Poisson distribution, the exponential distribution,
etc.

For a random variable X, if don't know its distribution,
then can't say what the numerical value of its
standard distribution is.

Moreover if for some random variable X
that has a standard deviation want to know, say,
the probability

P( -5σ <= X <= 5σ)

then that is an area and need the distribution
of X to find the numerical value.

All that is 100%, completely, totally, absolutely
true. That's what σ or standard deviation is.

In particular, the statement

"In statistics, σ refers to an area under the normal distribution defined in terms of standard deviations (1σ = 1 standard deviation)."

is flatly false. The field of statistics has no
such statement or convention.

Yes, if want to be really sloppy, make some
assumptions not clearly stated,
and have some conventions for identifying
some things in special ways, e.g., that
sigma is an area, then can do so. Maybe
some parts of physics do this. I do remember
when I was studying physics the prof handed
out a little book on how errors were
handled in physics. The book was a sloppy
mess and one of the reasons I lost respect
for accuracy and precision in physics and
majored in math instead.

It is true that about 100 years ago some
fields of study, especially parts of psychology
and much of education, concluded that the Gaussian
distribution was some universal law of data
handed down
by God. Well, God did no such thing. Still,
some people in educational statistics believe
that student test scores should have a Gaussian
distribution and, if the scores do not have
such a distribution, will, from many such
scores, find the empirical distribution and, then,
transform the scores so that the distribution
is closely Gaussian.

Maybe physics drank that Kool Aid that all
experimental errors of course, as given by
God, have a Gaussian distribution, that is
"a perfect bell curve",
and, then, yes, can get

"68% of the data is within one standard deviation of the mean, 95% is within two, and so on.",

and that σ has a particular numerical value
and regard standard deviation as an area.
Yes, maybe this is physics but it
is very sloppy thinking and
not mathematics, probability, statistics,
or anything from God.

Of course, even with a Gaussian assumption,
standard deviation does not have a
particular numerical value. Instead,
for Gaussian random variable X with
E[X} = 0 each of

P( -σ <= X <= σ)

P( -2σ <= X <= 2σ)

P( -5σ <= X <= 5σ)

has a particular numerical value.

Sorry 'bout that.