A brief translation for any confused American readers: in UK English, “State School” refers to what we in the US call a “Public School.” In UK English, “Public School” refers to what we in the US call a “Private School” (it is privately owned and/or run but open to the ‘public’). I think OP did a good job avoiding any possible confusion but this is a common enough error that it may be worth stating explicitly.
A brief translation for any confused American readers: in UK English, “State School” refers to what we in the US call a “Public School.” In UK English, “Public School” refers to what we in the US call a “Private School” (it is privately owned and/or run but open to the ‘public’). I think OP did a good job avoiding any possible confusion but this is a common enough error that it may be worth stating explicitly.