Said court shall, within the territorial limits of said city, have and exercise jurisdiction as follows:
It shall have and exercise exclusive original jurisdiction of all proceedings of a criminal nature for the violation of any ordinance of said city, and of every action of a civil nature for the enforcement of a penalty, or the recovery of a penalty or forfeiture imposed by any ordinance of said city for violation thereof, or for neglect to perform any duty by any ordinance imposed, and of every action for the collection of any license, fine, or penalty due from any person to said city, and required to be paid, or which is due and collectible under the ordinance[s] of said city.