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.