Im studying a Support Vectors Machine and for soft margin I found minimization problem in form like this:
Minimization problem

And this this formula seems pretty understandable where we have to minimize the weights and the consequence of this will be an increase of margin, but because it’s soft margin we allow some points to be at the wrong side of hyperplane, which represented by sum of slack variables, but after in sklean I found another…form of it?
Another form of minimization problem

And here we have loss and penalty functions and I cant really understand how can we interpret this, in other tutorials I saw it with slack variable and it was easier to me.
But now in my coursework I need to describe all of parametres of SVC from sklearn package, can you suggest some tutorials with it and which of two above formulas should I consider as the.. better one?
How can I deal with loss and penalty and what is the mathematical meaning of it?
Or which of the formulas should I rely on more?

I tried to find any understandable tutorials on this topic but I dont succeed much in it.