I agree that the definition is problematic, though commonly accepted. The argument for goes back to Hobbes, and thus is in one sense foundational to western democracies.
But it runs into problems when you look at the other aspect of western democracies, which is that power resides in the people and is only conditional granted to authority figures (i.e the "king" rules by consent of the governed, not by divine right). Thus the people have the ultimate right to legitimate violence.
Also remembering that Hobbes was arguing for a king, not an elected representative...
While it resembles Hobbes description of the Leviathan, and is in some sense a product of it, the definition is more directly product of fairly modern political science seeking to systematically define it's scope of analysis and the things it is addressing in ways that don't apply only to a narrow cultural and historical context than it does to Hobbes’ normative arguments.
> Thus the people have the ultimate right to legitimate violence.
That it makes a genuinely democratic state also one in which the state or government is the people is more a feature than a bug of the definition. (Though whether actual modern Western regimes are genuinely democratic is a question for which analysis of particular states has frequently suggested a negative answer.)
That argument might come from Hobbes but it wasn't his definition of a government either. Absolute authority as he saw it was simply a requirement for effective government.
Boxers have something of a monopoly on legitimate violence in the context of a boxing ring, but that doesn't make them a government.
But it runs into problems when you look at the other aspect of western democracies, which is that power resides in the people and is only conditional granted to authority figures (i.e the "king" rules by consent of the governed, not by divine right). Thus the people have the ultimate right to legitimate violence.
Also remembering that Hobbes was arguing for a king, not an elected representative...