Does it matter whether we put regularization parameter ($C$) with error or weight term in Kernel ridge regression?
Both formulations lead to the same solution if you correctly choose $C$ for both cost functions and if $C&gt;0$.

If we have the regularized loss

$$J_1=\dfrac{1}{2}\sum_{n=1}^Ne_n^2+\dfrac{1}{2}C\sum_{k=0}^pw_k^2$$

we will have strong regularization for larger $C$ and small regularization for small positive $C$.

If we devide the loss $J_1$ by the positive $C$ we obtain the loss

$$J_2= \dfrac{1}{2C}\sum_{n=1}^Ne_n^2+\dfrac{1}{2}\sum_{k=0}^pw_k^2.$$

As $C&gt;0$ we just scaled our loss funtion $J_1$, hence the minimum will not change. But the interpretation of $C$ will change if we replace it with its inverse as proposed in your question to obtain the loss

$$J_3= \dfrac{1}{2}\tilde{C}\sum_{n=1}^Ne_n^2+\dfrac{1}{2}\sum_{k=0}^pw_k^2.$$

With $\tilde{C}$ very small we will have strong regularization. And for large $\tilde{C}$ we will have small regularization. That is the reason why we sometimes call $\tilde{C}$ the inverse regularization parameter (e.g. see support vector machines).