Mr. Biden told a U.S. Conference of Mayors meeting in Washington that it was “pure coincidence” that he was in Lancaster County, Pa., on Oct. 2, 2006, when a deranged gunman was killing students nearby at a tiny Amish school. “I happened to be literally — probably, it turned out, to be a quarter of a mile [away] at an outing when I heard gunshots in the woods,” Mr. Biden recounted. “We didn’t know. … We thought they were hunters.”