Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian

The Product of Two Gaussian Random Variables is not Gaussian distributed:

But the product of two Gaussian PDFs is a Gaussian PDF:

What is going on here?

What am I doing when I take the product of two pdfs
vs. when I take the product of two variables from the pdfs?

When (what physical situation) is described by one,
and what by the other?
(I think a few real world examples would clear things up for me)

The product of the PDFs of two random variables $X$ and $Y$ will give the joint distribution of the vector-valued random variable $(X,Y)$ in the case that $X$ and $Y$ are independent. Therefore, if $X$ and $Y$ are normally distributed independent random variables, the product of their PDFs is bivariate normal with zero correlation.
On the other hand, even in the case that $X$ and $Y$ are IID standard normal random variables, their product is not itself normal, as the links you provide show. The product of $X$ and $Y$ is a scalar-valued random variable, not a vector-valued one as in the above case.