The MTA is as you say a corporation, but it is a publicly (or stte owned) owned corporation. By itself, it is not simply an agency of government. Further more, it is a HOLDING COMPANY, which wholly owns other corporations, such as the LONG ISLAND RAIL ROAD, The METRO NORTH COMMUTER RAILROAD, and it also manages city and state properties such as NEW YORK CITY TRANSIT AUTHORITY, and the TRIBORO BRIDGE AND TUNNEL AUTHORITY.
LIRR and MNCR are still corporations in their own right, they hire and dismiss employees and they are not public employees.
If we could make it any more complicated, we surely would.