$$\rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\text{SD}(X) \text{SD}(Y)}.$$
This is for any $X$, $Y$, be they distributed Gaussian, Poisson, Slash, Gumbel, Wishart provided the moments defined above actually exist.
Whether or not this is true is not a democratic decision. The above is the definition you will find in any textbook or course.
You should write to the author and ask them to verify, or at least expound on the idea. I'm sure what the author meant to say is that the Pearson correlation is not a straightforward or meaningful summary if the data are not Gaussian (even then, a contentious point). Otherwise this is an interesting finding and the author should definitely submit their findings to JASA if it can survive a round of peer review!
Specifically, Jason says
It is easy to calculate and interpret when both variables have a well understood Gaussian distribution.
which is actually true, but then he continues:
When we do not know the distribution of the variables, we must use nonparametric rank correlation methods.
which is something worse than false, it's replacing careful thought and data analysis with a knee jerk reflex. A better answer would be,
When we do not know the distribution of the variables, we must carefully inspect scatter plots, and the shape of smoothed estimates of the mean response relating $X$ and $Y$, and the constant or heteroskedastic variance structure around the mean, considering a family of parametric or non-parametric summaries that might relate two variables. The Pearson correlation may in fact be a reasonable summary of the first order trend in the data, and confidence intervals and statistical tests may be well calibrated if the sample size is reasonably large, otherwise robust error estimates such as the jackknife may be employed to obtain summaries that are better calibrated.
But jeez who wants to read all that?
