"Public Corporation" can refer to a corporation that offers
shares on the public market, or it can refer to a corporation that
is owned by the government. Similarly, "Private Corporation" can
refer to any corporation that is not government owned, or a
corporation that does not list its shares on public share markets.
A "crown corporation" is one that is owned by the government. This
term is more widely used in UK, some Commonwealth countries and
other monarchies.