SVC vs LinearSVC in scikit learn: difference of loss function

According to this post, SVC and LinearSVC in scikit learn are very different. But when reading the official scikit learn documentation, it is not that clear.

Especially for the loss functions, it seems that there is an equivalence:

And this post says that le loss functions are different:

• SVC : 1/2||w||^2 + C SUM xi_i
• LinearSVC: 1/2||[w b]||^2 + C SUM xi_i

It seems that in the case of LinearSVC, the intercept is regularized, but the official documentation says otherwise.

Does anyone have more information? Thank you

from SVC vs LinearSVC in scikit learn: difference of loss function