"Monopoly on violence in a given non trivially sized area"

That is the standard libertarian definition. It fits every state to ever exist. It reflects the crucial difference between an organized and/or cooperating group of people and a state. It makes sense and works.

What is your problem with it exactly?

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This was also my understanding of 'state', I didn't know it was a libertarian definition. What other definitions are commonly used?

That's the same definition in statistspeak (trying to legitimize state violence) and wrongly ignoring issue of physical location, which is in fact necessary.

Enforcement of the monopoly on violence always is delimited by an area where thst monopoly applies. Even if it was the entire universe you must specify that it's in the entire universe.

Aside from some wording issues. (statists often try to legitimize the stste in definition) this one is pretty widely accepted.

If you disagree with both of it's main points, maybe you should consider calling whatever it is you have in mind a different word.

What is the essence of state in your view?

That is not a definition. That's random words.

Okay, nevermind.

Also it doesn't matter whether you are or aren't libertarian. You accused libertarians of crafting defintions ad hoc and then misusing those definitions. (also for a reason that is likely just a stupid insult instead of an honest criticism)

If you want to just throw feces, be my guest, if you truly think there is something wrong with the libertarian attitude towards definitions and the state, I highly suggest stepping up your game.