What is the more general term for a lieutenant or captain in charge of a single apparatus and the firefighters assigned to it on a given shift?

The term that most accurately describes an individual who is in charge of a single piece of firefighting apparatus and the firefighters assigned to that unit during a specific shift is "company officer." A company officer is typically responsible for supervising the crew, ensuring compliance with procedures, and managing operations at the scene of an emergency. This role is foundational within the fire service, as it directly involves leadership and accountability for a team.

In contrast, a platoon leader usually oversees multiple companies or units within a larger organizational structure, focusing on broader operational strategies rather than individual apparatus. Meanwhile, a squad authority wielder, while not a standard term within the fire service, implies a position of leadership within a squad but does not encapsulate the broader responsibilities of a company officer. Lastly, a deputy assistant chief is a higher-ranking position that oversees larger operations or multiple companies, distancing themselves from the day-to-day management of a single apparatus.