(a) In an election ordered by the commissioners court, county judge, county board of school trustees, or any other county authority and not held at county expense, the authority ordering the election shall appoint the early voting clerk.
(b) If the county clerk is appointed as early voting clerk under this section, the county clerk shall serve in that capacity, and the authority responsible for paying the expenses of the election shall reimburse the county for the time spent by the county clerk as the early voting clerk and by the county clerk's deputies as deputy early voting clerks.
(c) To be eligible for appointment as early voting clerk under this section, a person other than the county clerk must meet the requirements for eligibility for service as a presiding election judge, except that the appointee must be a qualified voter of the county and is not required to be a qualified voter of any other particular territory.