The president’s contempt for Mr. Trump took on a personal dimension as well when he recalled his grandparents from Kansas and said, “I don’t know if they had their birth certificates” — a reference to Mr. Trump’s leadership of the so-called birther movement that raised questions about Mr. Obama’s citizenship.

Wednesday signaled a transition for the party. Emotion suffused the convention hall: Some delegates, in tears, were not ready to say goodbye to Mr. Obama yet, and others — particularly some liberals and young Democrats — were not ready to accept Mrs. Clinton as their new leader. As she prepares to give her nomination acceptance speech on Thursday night, the left wing of the party still remains divided, while many Republicans appear ready to fall in line behind Mr. Trump.

Mr. Obama’s speech, a passionate defense of Mrs. Clinton’s vision and character, did not itself herald the start of new political era. Mrs. Clinton has wrapped herself in the cloth of the Obama presidency rather than break with him and offer a new path, like Vice President George Bush’s promise of “a kinder, gentler nation” in 1988 after the Reagan years.

Instead, the lineup on Wednesday reflected a party attempting to rally its own partisans and attract Republicans with blunt warnings that, whatever they may think about the new Democratic standard-bearer, they must all do their duty to thwart Mr. Trump.