Is everyone in a fleet technically part of the admiral's staff?
Short answerNo; the fleet is the force being commanded, while the admiral has a dedicated staff to coordinate it.
NotesThe fleet was the force being commanded, while the admiral's staff was the smaller team coordinating its operations. A Giuseppe Garibaldi-class cruiser carried 555 people normally and 578 as a flagship. The difference makes a twenty-three-person staff visible in the complement figures. It also shows how compact an early twentieth-century flag organisation could be without confusing it with the force being commanded.