Whenever a person, committed to the State Department of State Hospitals or the State Department of Developmental Services under one of the commitment laws providing for no reimbursement for care and treatment to the state by the county of commitment, is transferred under Section 6700 to an institution under the jurisdiction of the department where the county is required to reimburse the state for such care and treatment, the State Department of State Hospitals or the State Department of Developmental Services may have the original commitment vacated and a new commitment issued, designating the institution to which the person has been transferred, in order to make the county liable for the care and treatment of the committed person to the extent provided by Sections 7511

and 7512.