WASHINGTON — President Obama was backstage at an auditorium at George Washington University last April preparing to give a major speech, when William M. Daley, then his chief of staff, spied an unexpected guest in the audience: Representative Paul D. Ryan, the Republican chairman of the House Budget Committee, whose budget plan Mr. Obama was about to shred.

“Try to tell the president!” Mr. Daley directed an aide.

It was too late to deliver a warning. Mr. Obama went on stage and outlined his proposal to reduce deficits — but not before he flayed the Ryan plan, saying its deep tax cuts and deeper spending reductions would harm students, seniors, the disabled and the nation.

“It’s not going to happen as long as I’m president,” Mr. Obama vowed.

Ten months later, the attack that left Mr. Ryan fuming in the front row is better remembered than the ideas Mr. Obama presented that day, administration supporters lament.

It came just a few months after the president had opted not to endorse the recommendations of a deficit commission he had created in hopes of brokering a bold, bipartisan deficit deal. That gave rise to a portrayal that has stuck, popularized by Republicans, pundits and some Democrats: that the president, out of political timidity, snubbed his own panel’s plan.