What are the terms for lawmakers at the city and county level?

Local authorities.

Explanation:

The most local legislative branch is the local council of a town or a county. They are very limited in power. Another local legislative branch is county commision, which are represented by the County Comission members.

A state legislature is the lowest level of the legislature, which transferred most of its power to state or federative bodies. Even though, federal government derives its power from the state and local bodies, state and local bodies give that power to the federal government through its representatives and lawmakers.