Liability and the collection and enforcement of such liability for the care, maintenance, services, and supplies specified in this section are governed exclusively by this section, except in cases of child support ordered by a court under s. 48.355 (2) (b) 4. or (4g) (a), 48.357 (5m) (a), 48.363 (2), 938.183 (4), 938.355 (2) (b) 4., 938.357 (5m) (a), or 938.363 (2) or ch. 767.