Earlier on Friday, the president had told senior aides that he had decided to remove Mr. Bannon, according to two administration officials briefed on the discussion. But a person close to Mr. Bannon insisted that the parting of ways was his idea, and that he had submitted his resignation to the president on Aug. 7, to be announced at the start of this week. But the move was delayed after the racial unrest in Charlottesville, Va.