WASHINGTON — If Vice President Joseph R. Biden Jr. decides to run for the presidency again, his best chance may well be to present himself as President Obama’s third-term successor. On Tuesday, Mr. Biden took the first step, describing himself as Mr. Obama’s most essential partner while taking subtle swipes at his would-be rival for the Democratic nomination, Hillary Rodham Clinton.

Along the way, Mr. Biden sought to recast his role in the raid to kill Osama bin Laden, arguably the most picked-over moment of Mr. Obama’s presidency and one that might hurt Mr. Biden’s presidential chances.

Mr. Biden had previously said that he had advised the president against launching the special forces raid on the Abbottabad compound where Bin Laden was suspected of hiding. At a Democratic congressional retreat in January 2012, Mr. Biden said that almost every other official in the Situation Room had hedged on a response when asked by the president whether he should order the raid.

“I said, ‘We owe the man a direct answer. Mr. President, my suggestion is, don’t go. We have to do two more things to see if he’s there,’ ” Mr. Biden said then.