What is a "variable index" in the Gaussian perspective?
The motivation behind this diagram is to explain how the 2 dimensions of a 2D Gaussian are related to each other depending on the covariance matrix. The author of the article refers to the dimension as 'variable index'. By reading further in the article, the author explains that as the off-diagonal element of the covariance matrix is 0.9, the 2 dimensions are highly correlated. So if we sample a point from this 2D Gaussian, the first variable-index or dimension of the sample will have almost a similar value to the second dimension (=2) in the example. On the other hand, if the off-diagonal element is 0.0 then the two variable-indices or dimensions will be very different.

This example builds up to the idea of defining a kernel function in the Gaussian Process.