Questions on the Generalized Tikhonov Regularization
One way to interpret the Tikhonov Regularization is using the Maximum A Posteriori (MAP) framework.
Lets' say we have a model of the form:
$$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$
Where $ \boldsymbol{n} \sim N \left( 0, {\sigma}_{n}^{2} \right) $, namely Additive White Gaussian Noise, and the prior knowledge about $ \boldsymbol{x} $ is $ \boldsymbol{x} \sim N \left( 0, {\sigma}_{x}^{2} \right) $.
Then the MAP estimator is given by:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2 {\sigma}_{n}^{2}} {\left\| H \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{1}{2 {\sigma}_{x}^{2}} {\left\| x \right\|}_{2}^{2} $$
So basically Tikhonov Regularization arises when we have prior knowledge or can model the parameters to have some kind of Gaussian Distribution.
Sometimes, our model is about the derivative (Very popular in Imaging). So we can say the derivative is distributed by Gaussian Distribution. Another way is having different covariance matrix (As in your example).