One day last summer, Gene Mulak, observing carnage in the sand trap, decided it was time to rescue the Leader of the Free World. "Open the clubface more!" the golf pro yelled to President Barack Obama. A rank of bodyguards stiffened when they heard the shouting, but the commander-in-chief continued to hack away, sand flying, recalls Mr. Mulak, a resident professional at the Vineyard Golf Club in Martha's Vineyard, Mass. Mr. Obama waved off the guards and welcomed Mr. Mulak into the pit. The pro says he gave the president tips on his golf stance and his swing, both of...