One could make an argument that the state does not hold a monopoly on violence. It holds a monopoly on the legitimization of violence. You're allowed to use deadly force to protect yourself from death or serious bodily harm in the majority of jurisdictions, provided it is not disproportionate. What matters is that the state declines to prosecute this, but does prosecute the person who hypothetically shoots someone on Fifth Avenue for no reason.
That's just semantics, for all intents and purposes the monopoly on violence is exactly that: the monopoly on the legitimisation of violence, only the state allows violence, you can only perform a violent act under the provisions the state allows, hence it's de facto and de jure holding the monopoly on violence.
Stating this because I'm not sure what differentiation you are trying to achieve with the semantics game, the meaning is the same for any interpretation.
I am specifically not arguing the state has a monopoly on violence - as with the GP, people are imputing arguments I’m not making. It’s the fact that the state cannot have a monopoly on violence that forces the state to be powerful enough that its threat of violence outweighs others (and, bluntly, defines what the state IS, for all practical purposes: it is the entity whose threats of violence supersede all others.)
