Nah, you have it wrong re:Westminster - several of my friends & my ex attended, and it most decidely is a public school; highest Oxbrige acceptance of any school in the UK; the only oddity, understandable because it's in central London, is that boarders are only one fourth of the intake. (Same with Godolphin & Latymer & with St Paul's for the same reasons.)

... just looked it up; est. 1179, refounded 1560; one of the nine schools in the Public Schools Act of 1868 (Charterhouse, Eton, Harrow, Merchant Taylors' School, Rugby, Shrewsbury, St Paul's, Westminster, Winchester.)