Again, you are excluding the critical phrase
legitimate claim, which is at the heart of Weber's definition. That is, the right and legitimacy of that right, is restricted to the state, or an entity acting in the effective capacity of a state, whatever it happens to call itself.
Absent this, one of three conditions exist:
1. There is no monopoly. In which case violence is widespread, and there is no state.
2. There is no legitimacy. In which case violence is capricious.
3. Some non-state power or agent assumes the monopoly on legitimate violence. In which case it becomes, by definition The State.
You might want to consider what a "state" which lacks a monopoly on the legitimate claim to the monopoly on force would look like. To what other entity would it cede that legitimate claim, and/or how would it prevent other entities from enacting capricious violence, as has occurred from time to time in the world, and even now.
The state's claim is to legitimacy. A capricious exercise would be an abrogation of legitimacy
Weber, Max (1978). Roth, Guenther; Wittich, Claus (eds.). Economy and Society. Berkeley: U. California Press. p. 54.
https://archive.org/details/economysociety00webe/page/54/mod...>
The "monopoly on violence" or "monopoly on force" short-hands are a much more recent emergence, and seem to originate with Murray Rothbard (1960s) and Robert Nozick (1970s), though widespread usage of that phrase really only begins to take off after 1980, per Google's Ngram Viewer: https://books.google.com/ngrams/graph?content=monopoly+on+vi...>
That shorthand has become quite popular, and is often cited by Libertarians as key to their adopting that particular ideology.[1] As expressed by them the formulation is both incorrect and misleading.
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Notes:
1. E.g., Penn Jillette, https://www.newsweek.com/penn-jillette-how-became-libertaria...> and Charles Koch https://www.newsweek.com/penn-jillette-how-became-libertaria...>.