(1) An action against a sheriff, coroner or constable upon a liability incurred by the doing of an act in his official capacity and in virtue of his office or by the omission of an official duty, including the nonpayment of money collected upon an execution, subject to the provisions of § 15-3-560; and

(2) An action upon a statute for a penalty or forfeiture when the action is given to the party aggrieved or to such party and the State, except when the statute imposing it prescribes a different limitation.