So drop the word 'exclusive'.
In our high school we had a student government. We had no prisons and no guns.
And the 'government' had no authority without the city and such other systems (ie government) ok. Also, if the students got out of hand they can be thrown out of school or otherwise punished. Authority in the sense of government regulation (if the word is used correctly) means that the persons assuming the authority can impose actions upon you and your property WITHOUT your permission AND use force if necessary. A police officer caries that gun on their hip because they are saying EXPLICITLY that they will shoot you dead if provoked enough. Provoking a authority figure means not doing what they want done the way they want it done.
You can't simply take over the ownership of words in the English language. 'Government' is indeed the name used for an entity that exercises authority or enforces laws. There can be more than one government exercising control over the same geographical or political area, and that control need not be effected with guns.
Yes, there can be more than one governmetn and each of them is in a heirarchy which EXPLICITLY defines what their individual jobs are. Should the states get out of hand you can bet the feds will send in big guns. You strike me as a person who trusts governments.