Usually people say "private corporation" to mean any company that is unaffiliated with the government, or any company run mainly for profit. It's impossible to buck this trend, so corporations with no issues of stock, or negligible amounts of stock held by the public, are called "private companies" despite the fact that not all companies are corporations. Cargill, Inc., for example, has annual revenues of $117 billion, but no shares available for trade. On the other hand, PricewaterhouseCoopers and Ernst & Young are private companies but not a corporations. They are limited liability partnerships. The term "closely held" would not apply to them since partnerships do not issue stock.
Publix Super Markets (at the other end of the spectrum) is closely held--by its 125,000 employees.