The correct, proper, and original formulation by Max Weber is the monopoly on the legitimate use of violence.
The alternatives are either granting all parties equal right to any chosen violence they choose to enact, granting the right to some particular entity, or attempting ... and by what means specifically without a state and sanction of legitimate violence ... to bar all use of violence.
Note too: legitimacy does not mean unlimited or capricious. It does, however, mean that the sanction of violence originates from, and is enacted either directly by, or through licence (e.g., self-defence) from, the state.
Misuse of this term is rampant, particularly among so-called Libertarians.
"The monopoly of the legitimate use of physical force, also known as the monopoly on violence"
I think the "legitimate" is implied - a State is a thing that creates Legitimacy and enforces it through Violence. To supplement this it monopolizes Violence by inflicting Violence on those who commit it without Legitimacy.
The alternatives are either granting all parties equal right to any chosen violence they choose to enact, granting the right to some particular entity, or attempting ... and by what means specifically without a state and sanction of legitimate violence ... to bar all use of violence.
Note too: legitimacy does not mean unlimited or capricious. It does, however, mean that the sanction of violence originates from, and is enacted either directly by, or through licence (e.g., self-defence) from, the state.
Misuse of this term is rampant, particularly among so-called Libertarians.