WASHINGTON — President Trump sat forward on the edge of his chair and chatted at length with reporters in the Oval Office on Thursday, unbowed but for him a little subdued. The day after he was impeached for high crimes and misdemeanors, he dismissed the judgment of the House of Representatives and punched back by celebrating with a Democratic congressman who switched parties to stand with him.

“I don’t feel like I’m being impeached because it’s a hoax, it’s a setup,” Mr. Trump insisted as he showcased Representative Jeff Van Drew of New Jersey, the newly minted Republican, and looked beyond his seemingly certain acquittal in a Senate trial to next fall’s election.

“I’m beating everybody by a lot,” the president said, “and I think that’s where we’re going.”

For Mr. Trump, it was the first day of his new reality, the first day when he woke up with the scarlet letter of impeachment marked with indelible ink on his page in the history books. No matter what else happens, he now enters posterity as the third president to be impeached.

But barring the unforeseen, he will also be the first impeached president to face re-election, setting up a 320-day campaign to convince voters that he was right and his accusers were wrong. He has a chance that Andrew Johnson and Bill Clinton never had — to mitigate, at least, the sting of impeachment — and his political operation wasted little time mounting a counterassault on what Mr. Trump characterizes as the corrupt, liberal Democrats who orchestrated a largely party-line scheme to nullify his election.