Actually, impeachment requires Senate and then Congress to say 'go for it, impreach the loser' but, the Supreme Court has to actually impeach him. Clinton only got through Senate, the court didn't hear it.
That is not true. The House of Representatives votes on Articles of Impeachment. If passed by a simple majority, the Senate then tries the President (or others). The trial is presided over by the Chief Justice of the Supreme Court, or the Vice President as President Pro Tempore of the House.
If the Seante votes with a two-thirds majority to approve the impeachment, then the President has in fact been impeached.