BEIJING—President Donald Trump blended tough talk with chumminess at a U.S.-China summit that emphasized personal rapport over progress on deep-seated differences, calling Xi Jinping “a very special man.”

The two countries’ rivalry will be more clearly on display as the two presidents move on to Vietnam on Friday and outline conflicting visions for Asia’s future to regional leaders. Mr. Trump is promoting a “free and open” region to counter the Chinese president’s drive to make his country its dominant power.

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