Within fifteen days after the service of the notice of the appeal the respondent may serve upon the appellant and the magistrate an offer in writing to allow the judgment to be corrected in any of the particulars mentioned in the notice of appeal. The appellant may thereupon, and within five days thereafter, file with the magistrate a written acceptance of such offer, and in such case the magistrate shall thereupon make a minute of such acceptance in his docket and correct the judgment accordingly, and the judgment, so corrected, shall stand as his judgment and be enforced accordingly; and any execution which has been issued upon the judgment appealed from shall be amended by the magistrate to correspond with the amended judgment. If the offer be made and accepted by the appellant, the appellant shall recover all his disbursements on appeal and all his costs in the court below.