Public institution means an institution that is the responsibility of a governmental unit or over which a governmental unit exercises administrative control. The term public institution does not include -
(a) A medical institution as defined in this section;
(b) An intermediate care facility as defined in §§ 440.140 and 440.150 of this chapter;
(c) A publicly operated community residence that serves no more than 16 residents, as defined in this section; or
(d) A child-care institution as defined in this section with respect to -
(1) Children for whom foster care maintenance payments are made under title IV-E of the Act; and
(2) Children receiving AFDC - foster care under title IV-A of the Act.