“It’s game on, the beginning of the general election,” said Representative Ann McLane Kuster of New Hampshire, the chair of the New Democrat Coalition, a group of 97 centrist House Democrats. “We’ve got to win this.”

In a race without historical parallel — a contest between two presidents, one of them facing 91 criminal charges — Mr. Biden is making an extraordinary gamble, betting that Mr. Trump remains such an animating force in American life that the nation’s current leader can turn the 2024 election into a referendum not on himself but on his predecessor.