# Is a kernel a correlation or a covariance function?

I am reading this paper on multi-fidelity optimization, where I came across an introductory section on kriging a.k.a. Gaussian Process regression (see Figure below). It confused me about the notion of the kernel and whether it pertains to correlation (what seems to be implied by defining $\psi$ in the paper as a similarity measure) or to covariance.

People like Duvenaud have defined a kernel including the variance $\sigma^2$ in front of the exponential defining similarity. People like Rasmussen (PDF page 5) define the kernel without the variance in front.

From the definition that correlation is covariance normalized by variance (i.e. $\textrm{cov} = \textrm{corr}\cdot \sigma^2$) we could say that Duvenaud denotes the covariance as kernel whereas Rasmussen denotes the correlation as kernel. Confusingly enough, Rasmussen uses the term 'kernel' and 'covariance function' synonymously (as does Wikipedia, and SE).

Can anybody make sense of this confusing vocabulary?