how do other companies define non-people managers? In other words, do you use the manager title if an individual does not manage people? If so, what determines when you use the manager title if the role doesn't manage people?

Manager title might be used when managing projects or programs, and not necessarily people. Normally the description of the program or project would come first. [Insert program name] Manager.  Although it is the work that needs to be managed, not people, so Team Leader may be more appropriate.

A manager can manage either people or a function, program or project so they don't need to have direct reports. But, if you want the manager title to signify direct reports then you could use the term "Lead", "Subject Matter Expert" or something else appropriate to your organisation for non-people managers.