The mass shooting in Orlando, Fla., elevated an issue that Republican Donald Trump has been stressing throughout his presidential campaign: the threat posed by jihadists bent on killing Americans at home.

Hillary Clinton, his Democratic opponent, on Sunday emphasized other dynamics. She said the U.S. must work with “allies and partners” to foil terrorists. She also made note of the gay and lesbian targets of the killings, denouncing “an act of hate” against a community that is part of the political coalition she hopes will...