The stock market is an example of Public finance even though an individual or a business investment in it is a "private investment".
And then there is the third category that you missed entirely,
This is characterized by bookkeeping that would be illegal in any other form of finance. Absolute corruption, theft, and favoritism. It usually involves having several sets of books and statistics designed to thwart investigation. It is financed by flat out theft from many and supported by those given power and trust that deserve neither.
and take their power away.