private citizens were operating in way that most reasonable people would call authoritarian, but by your definition cannot be called authoritarian because it's only authoritarian when the state does it.

The state absolutely played a roll in using violence to maintain slavery.

Is it possible for a state to be authoritarian through inaction? Suppose, for example, interracial relationships are technically legal, but every time one happens or is even suspected, a lynch mob strings someone up on a tree, and the government fails to prosecute.

This is an interesting point. I think strictly it doesn’t “count” but if you consider this behavior as playing a roll for the state, I could see this counting.

However, again, I think this is missing the point. Something can be despicably violent whether or not it is specifically “authoritarian”.

But if those people pose a genuine threat to others, then doesn't the state have an obligation to stop them in order to not be authoritarian

If it’s a threat to other people yes the state should intervene. If it’s a threat to the political status quo without otherwise needing the government to step in then it’s “authoritarian”. It can be an abuse of power either way.

And for that matter, isn't it authoritarian for the US to allow Coca-Cola to fund death squads, in the original example?

So I’ll admit I had to look this one up. From what I can tell, the stance of the US in that case that it was Colombia’s job to prevent or prosecute crime occurring in its jurisdiction. Personally I do wish big companies would face international consequences more often.

Applying the definition, I think you could consider this an example of Colombian “authoritarian violence”.