(1):

In the United States, a free primary, grammar, or high school maintained by the local government.

(2):

In Great Britain, any of various schools maintained by the community, wholly or partly under public control, or maintained largely by endowment and not carried on chiefly for profit; specif., and commonly, any of various select and usually expensive endowed schools which give a liberal modern education or prepare pupils for the universities. Eton, Harrow, Rugby, and Winchester are of this class.