You are confusing pressing charges and indictment. Pressing charges just means you accuse somebody of a crime and “press” the prosecutor to indict them. So the state does have the ultimate say on who is prosecuted, but that doesn’t mean you can’t press charges.

Many countries do have the concept of private criminal prosecutions.

As far as I am aware it isn't a specific thing, but a general catchall term for going through the process of filing a criminal complaint, and seeing it through to completion. Maybe there is better words for it but "pressing charges" is what they use on TV so it is top of mind.

In general I meant there is a difference between criminal and civil law, and suing generally refers to civil not criminal law.

It is a thing. In America pressing charges is when you accuse somebody of a crime and ask a prosecutor to bring criminal charges against them.