(a) By Clerk. On the filing of a warrant, or the return, fully paid or satisfied, by the sheriff or other officer of any execution issued on any judgment, the clerk shall forthwith enter satisfaction on the record, provided, however, that satisfaction of a judgment in favor of a minor for more than $5000 shall not be entered except on court order, unless a guardian has been appointed for the minor's property or the minor has come of age. The clerk may enter satisfaction on the record as to any co-defendant filing a warrant stating that the party has paid the full share of the judgment even though a portion of the total judgment remains unsatisfied.
(b) By Order of Court. If a party receiving full satisfaction of a judgment fails to enter satisfaction on the record or deliver a warrant to satisfy, the court may on motion by the party making satisfaction, order satisfaction of the judgment to be entered of record.