What does "exempt" mean? Generally, it means that the employee is exempt from the overtime and rest/meal period provisions of California law. Nonexempt employees generally must be paid at least minimum wage, on an hourly basis, with overtime pay, and must be provided meal and rest periods within set times. Failure to do any of these things will result in penalties/premium pay being owed. Exempt employees are generally (but not always) paid on a salary basis that does not vary depending on hours worked.
It is important to note that paying someone a salary DOES NOT make them exempt. In fact, a significant amount of wage and hour lawsuits arise from situations in which an employee is paid a salary, but should have been paid premium pay for all overtime hours worked.
An agreement between the parties as to exempt status is not binding or even particularly meaningful. These rights cannot be waived. Whether someone is exempt or nonexempt depends on whether they meet the legal test for exemption.
Similarly, titles are not important. Calling someone a "manager" or "supervisor" does not make them exempt.
Under both state and federal law, the burden is on the employer to prove exempt status, and it is the job duties that determine if someone is exempt or not.