Should a manager refer to people who work under them as 'my employees' or 'my staff'?

It depends on the field. The higher the expectation from the team of buy-in, being heard and contributing to the team, the higher the expectations of mangers referring to the people who work for them as "my team." There is a push away from "my staff" as it implies a command and control style of leadership. "My employees" will also, and possibly more so, be received as a command and control. This may be more or less relevant based on the norms of your field and the culture of your organization.