Texas Election Code - ELEC § 66.001. General Custodian of Election Records

(1) the county clerk of each county wholly or partly situated in the territory covered by the election, for an election ordered by the governor or by a county authority or for a primary election;

(2) the city secretary, for an election ordered by a city authority;  and

(3) the secretary of the political subdivision's governing body or, if the governing body has no secretary, the governing body's presiding officer, for an election ordered by an authority of a political subdivision other than a county or city.