# Should a nugget ever shift the variogram away from zero at distance zero?

I had frequently seen the definition for a "rigorous" spatial isotropic semivariogram being defined as:

$$\gamma(h) = K(0) - K(h)$$

Where $$K$$ is a positive definite covariance matrix. If the nugget $$\sigma$$ is applied to the covariance function $$K$$, then it seems to me that it would always be that $$\gamma(0) = 0$$, but that $$\gamma(0+\delta)>\sigma$$ for all distances greater than zero.

However, often in practice a linear function is used to determine $$\gamma$$:

$$\gamma(h) = \sigma + \frac{h}{\beta}$$

Should I really define the nugget in terms of the covariance matrix, or should I make it a separate term:

$$\gamma(h) = \sigma + K(0) - K(h)$$

Am I making a mountain out of mole hill here? Are there any reasons to prefer one form or the other?

To visualize, is it more "correct" to pick one form over the other:

Based on the idea of nugget as allowing a non-zero offset at zero, I could see the former should be preferred: but this seems to imply that the nugget is no really part of the assumed covariance matrix.