MINNEAPOLIS — Hillary Rodham Clinton staked her claim on Friday to lead Democrats in 2016 and beyond, delivering a fiery speech to hundreds of party officials in which she attacked Donald J. Trump and other Republicans for “hateful” remarks — “The party of Lincoln has become the party of Trump,” she said acidly — and pledged to rebuild the Democratic political machine to help candidates win races nationwide.

But if Mrs. Clinton was seeking to unify Democrats behind her, two of her rivals for the nomination — Martin O’Malley, the former Maryland governor, and Senator Bernie Sanders of Vermont — were hardly willing to go along, as they used their speeches at the Democratic National Committee’s summer meeting to aim unusual broadsides at the party overtly and Mrs. Clinton implicitly.

In the fiercest speech of his candidacy so far, Mr. O’Malley condemned his party’s leadership for curtailing the number of primary debates to six and scheduling them at times when few people would see them. (He went even further in a news conference afterward when asked if the party had “rigged” the debate calendar to benefit Mrs. Clinton and lower the visibility of her rivals. “Yes, I think so,” he replied. “Don’t you?”)

Mr. O’Malley’s repeated demand for more debates drew standing ovations from the audience and scowls from the party chairwoman, Representative Debbie Wasserman Schultz of Florida, seated a few feet away. And if the cheering Democrats were a sign that party members were not ready to coalesce around Mrs. Clinton, the energetic reception for Mr. Sanders — and bursts of applause for his own criticisms of the party — confirmed that unity would have to wait.