My understanding is that legally, it's not part of the government. That makes it private, with any voluntary government relationships being just that. This doesn't make what you are saying untrue, but what you are saying isn't exactly clear either.

(Thought experiment: Obviously, if one did wield influence on a central bank after winning the rights to run it from a nation, one would try to keep public perceptions as wishy-washy as possible in order to keep pockets safely lined and one's fiscal hegemony in place.)