The White House did not respond to a request for comment about the apparent discrepancy between Musk’s directive and the guidance provided by agency heads. The scramble to discern Trump and Musk’s exact intentions with the email added another layer of uncertainty to an already-rattled federal workforce.

The F.B.I. director, Kash Patel, the director of national intelligence, Tulsi Gabbard, and others told employees not to respond to a directive from Mr. Musk to summarize their accomplishments.