For clarity:

Public schools - a handful of top level very expensive private schools, usually have existed for centuries, the children of the highest levels of society tend to go there, a great foundation for a career in public life. Examples include Eton, Harrow, Winchester, Marlborough.

Private schools - the rest of the private schools, less expensive, but less advantageous.

State schools - free, provided by the state for children in a local catchment area, this is where the rest of us go. Most are run by the local government, but there has been some experiments with 'foundation schools' which are tax payer funded but privately run.