The fact that governments can circumscribe the killings they do as “not murder” and have regular people believe them is the quintessential expression of the state monopoly on violence.

A monopoly on something doesn’t mean exclusive, 100% godlike control of it. It just means the legitimate provisioning of that thing falls under the authority of one entity. Copyright is a government granted monopoly on the right to copy a creative work but that doesn’t make it impossible for others to make copies, it just makes it illegal.

"The fact that governments can circumscribe the killings they do as “not murder” and have regular people believe them is the quintessential expression of the state monopoly on violence."

A state, and words and concepts, only exist insofar as people believe in them. You're only saying "murder is [defined]", not anything about governments. If you pretend you are making a statement about government or politics, it suggests that somehow a different system could achieve a better and different state. Everyone defining legal killing for themselves is not better or worse, it's a contradiction in terms - an absurdity.