(a) It shall be the duty of the justice of the peace or the clerk of the court rendering a judgment or sentence of conviction in misdemeanor cases when the defendant is committed to the county jail to file a copy of the judgment or sentence in the office of the clerk of the county court, together with a full itemized statement of the fine and costs adjudged, as well as a concise statement of the time the convict may be held to labor in discharge of the judgment or sentence.
(b) The clerk of the county court shall register the judgment or sentence in a book to be kept by him or her for that purpose.