The extravagant red-carpet welcome he received from President Xi Jinping in China, President Trump said, was one like no other American president had ever gotten. America’s relationship with President Rodrigo Duterte of the Philippines — accused of a bloody campaign of extrajudicial killings in his nation’s drug war — was now “better than ever,” he proclaimed. After a meeting with President Vladimir V. Putin of Russia, he said the two were developing “a very good feeling for each other” — despite what he later called efforts of “haters and fools” to drive them apart.