Correct me if I am wrong, but all the offender has to is pay a sum and walk away free?
No. Restitution means that the offender bears liability for the full consequences for what they did (read:
all of them), including the harm itself, long term effects, and enforcement costs. This can include long term or even lifelong obligations, restrictions, and supervision depending on the severity and the risk the offender poses.
Like, a fixed penalty system where you pay a predefined amount and you're done, that's what the state does. In a free society, liability reflects the harm you caused. If the harm done by an offender is ongoing or severe, the liability is ongoing or severe as well.
The reason why it's important to punish criminals (depending on the severity) is because imo that is the very definition of justice. The victims do not matter here. The suffering of the criminal is also secondary. What is important is that he dies. With the criminal being dead, he can no longer plague people and society with his atrocious actions, and their transgressions against other people are nullified.
Wow, removing the victim from "justice" entirely and replacing it with "kill the offender". I don't see how this preference for eliminating people you don't like qualifies as a theory of justice.
Killing the offender won't "nullify" shit. The harm already happened, nothing is restored, the victim is still harmed, and all you'd accomplish is adding another irreversible act. If anything, your plan would guarantee that nothing can ever be made right.
If your actual concern is preventing future harm, then say that. That would be a question of containment or exclusion, not a justification for arbitrary punishments like rape. Shit like that won't make the victim whole, it won't reflect any cost, and it won't solve the problem you're pointing at.
What is the meaning of [cost internalization]?
It means that the offender bears all the costs they imposed on others, instead of shifting them onto everyone else. The alternative would be that everyone else is effectively subsidizing aggression, just like you have in the current legal systems all over the world. In a libertarian society, proper liability makes aggression high-risk and unprofitable.
Call them whatever you want, you still need a distinction between justified and unjustified uses of force. If you reject that entirely, then there's no real difference between crime and punishment. All you'd have is a preference on who gets to do what.