The PGA Golf cup was begun with President Gerald Ford and is a match between US golfers and the rest of the world apart from Europe. The cup is named for the US Presidency in general, not any particular president.
(There are several Presidents Cups. If this wasn't the one you were looking for, please ask a more specific question.)
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It was created by the PGA. They decided that each year's honorary chairperson would be a current or president of one of the countries whose golfers participated in the tournament. Since inception in 1994, the honorary chairpersons has included the leaders of the US, the UK, Australia, Canada, and South Korea.