the simple or flavorful use of 'authority' usually just means someone is well respected on a topic "She's an authority on electrical motors" or that they posses some leadership qualities and are well liked.
the second and more dictionary definition of authority is authorization to rule. literally to dominate others because they are seen as legitimate.
your example of a club is a good test ground for these differences. it's up to you to run your club, make agreements with other members and share responsibilities even if people are happy to let you decide things or do most of the work. but your club doesn't have any claim to legitimacy, to true authority, because it doesn't seek to control ALL activities of that type. in the case of government, it very much does claim authority over the entire nation, to determine who can move where, who can work where and how, who must fight and who must be put to death.
there's no competition allowed with a structure like government. either obey or resist and face consequences. i'd much prefer a world of overlapping autonomous 'clubs' whose members decide for themselves than a world divided up by the greediest and most violent mafias.
anarchists don't have elections. anarchists have consensus and divide up or directly oppose each other if there's unresolvable differences.