>Who determines what the "truth" is?

A judge? You know, the very people we have collectively decided should be in charge of evaluating facts to understand the truth behind certain events and sanctioning those who broke the law as a consequence?

>You can't take every potential inaccurate or mean spirited statement to court

This is exactly what we do here in my country (Italy) as long as someone files a complaint, so your claim that this can't be done is false.

>How do you prove things that aren't really provable?

If guilt can't be proven beyond reasonable doubt, you're acquitted. It's a general principle.

>How do you prove if someone is lying or just uninformed?

It is part of a judge's job to analyze intent. It's also what, among many other things, allows a judge to discern between manslaughter or premeditated murder, for example.

>Now who reads, researched and then hands out punishments to the millions or billions of infractions each day?

As I briefly mentioned above, here in Italy you go to court only after a direct complaint (to the police) made by the person who feels like they have been offended by the statement in question (e.g. Hillary Clinton in your example). The police will only act on their own in case of speech that constitutes inciting discrimination or violence based on race, ethnic background, nationality or religion, of for spreading ideas related to racial superiority or to the negation of the historicity of the Shoah.