e. Service in any city or county institution that became a state institution on or before May eleventh, nineteen hundred twenty, only to the extent that such service is paid for by the state or by such institution.
Except as otherwise specifically provided in this article, service rendered on and after January first, nineteen hundred twenty-one, by a person entitled to retirement benefits for civil service employees pursuant to other laws, wholly or partly at the expense of the state or any political subdivision thereof, however, shall not constitute government service.