The Fed is a government agency in the same capacity Freddie and Fannie were government agencies prior to 2008 (post 2008, Freddie and Fannie are very much government controlled after the housing market debacle).
But to answer your question directly, true.
True because there is minimal government control of the Fed. Which separates it from a private ins ution like any size corporation or private bank.
False because the Fed has no legal requirement to the government save for board appointments.
So really, its both. They are quasi-government, a gray area. Gray areas do exist and the Fed is the best example of one.