DAVOS, Switzerland — President Trump swept into this moneyed Alpine village on Tuesday, full of brio and flattery, schmoozing with global business leaders as if there were no talk of removing him from office and no impeachment trial unfolding 4,000 miles away in Washington.

Mr. Trump appeared to relish the escape offered by the World Economic Forum and the friendly — to his face, at least — crowd of elites in the snow-covered Alps. He was in a jovial mood, according to people who spoke with him, engaging in animated conversations with chief executives like Brian Moynihan of Bank of America, Sundar Pichai of Alphabet and Marc Benioff of Salesforce.

He congratulated them on their companies’ stock performances and joked that he should have bought shares but that he had been forced to sell his holdings when he took office. As Mr. Trump and his family members darted among meetings in makeshift pavilions, they studiously avoided questions about the drama back home, where the Senate engaged in a fierce clash over the rules for putting the president on trial.

It was a day of two presidents. There was the stick-to-the-script Donald J. Trump riding high on a strong economy and representing the country on the international stage. And there was the Donald J. Trump under siege back home, depicted as an autocrat abusing the power of his office to take down domestic opponents and win re-election.