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

Question 1

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

Is the product of two Gaussian random variables also a Gaussian?
Also Wolfram Mathworld
So this is saying $X \sim N(\mu_1, \sigma_1^2)$ , $Y \sim N(\mu_2, \sigma_2^2)$ then $XY \sim W$ where W is some other distribution, that is not Gaussian

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

Calculate the product of two Gaussian PDF’s
Full Proof
This tutorial which I am trying to understand Writes: $N(\mu_1, \sigma_1^2)\times N(\mu_2, \sigma_2^2) = N(\frac{\sigma_1^2 \mu_2 + \sigma_2^2 \mu_1}{\sigma_1^2 + \sigma_2^2},\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}})$

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)

Question 2

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.

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

Answer

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