Great point. I think the distinction between public and private makes sense when talking money. "Public money" is usually in the context of government spending of money taken from the public, with an intended purpose to serve the public, while private money is profit made from business (albeit also from the public) and serves private interest and the business.

In the context of schools however it does get pretty muddy. In my country public schools are government-funded but completely free, but in states where even government-funded education is a business (USA, the UK etc) the distinction makes little sense really.