What does it mean when we talk about the "Gaussian likelihood" for a Gaussian Process? Is it true to think that the "Gaussian likelihood" only means we factor in a noise term into the covariance function? i.e.
$$cov(x,x') = K(x,x') + \sigma^2$$
In other words, we only add $\sigma^2$ to the diagonal of the covariance matrix. Is my understanding correct? Furthermore, what happens if the variance of the Gaussian likelihood is zero? Does the likelihood become non-Gaussian and we can't use the standard Gaussian Process inference any more?