It is useful to distinguish two types of independence for central banks: goal independence and instrument independence. If a central bank is free to set the final objectives for monetary policy, it has goal independence. If a central bank is free to choose the settings for its instruments in order to pursue its ultimate objectives, it has instrument independence.
Most central banks have specific legislative mandates and therefore do not have goal independence. Thus the “independence” of “independent” central banks is instrument independence under which the central bank has authority to choose settings for its instruments in order to pursue the objectives mandated by the legislature, without seeking permission from, or being overturned by, either the executive or the legislature.