All demands against a code city shall be presented and audited in accordance with such regulations as may be prescribed by charter or ordinance; and upon the allowance of a demand, the clerk shall draw a warrant upon the treasurer for it, which warrant shall be countersigned by the mayor, or such person as he or she may designate, and shall specify the fund from which it is to be paid; or, payment may be made by a bank check when authorized by the legislative body of the code city under authority granted by RCW 35A.40.020, which check shall bear the signatures of the officers designated by the legislative body as required signatories of checks of such city, and shall specify the fund from which it is to be paid.