You stayed it is run not for profit. Often public transport is run for profit by private companies. The profit is why they get involved. In many cases the private management is so far superior to f*cked up state incompetence, that they provide a better service and manage to turn a profit.
In the UK, almost the entire rail network was originally built and operated by the private sector.
So it is PUBLIC transport because the public may use it. Either run by a for profit company or some politicians.