For something I'm applying to, I need to know if I'm considered a student, a part-time employee, or a full-time employee.
The answer could heavily depend on the country you are located in.
The following is obvious:
- do you have a proper work contract with the university? If yes, then check your contract and see what kind of employee you are;
- are you officially enrolled at a university as a student? If yes, the program you are enrolled in tells you what kind of a student you are.
Now to the non-obvious part. As pointed out by others, in some countries you would be a either a full-time, or a part-time employee, but despite that, for many purposes (grant applications, student fellowships, conference registration, etc.) you could be considered a full-time student. This is the case for Germany (among others) where your contract would stipulate an amount of hours you are paid for, but you would be nevertheless expected to work full-time as a PhD. student and the university would have no problem issuing a certificate about your "studentship" for you. So to tackle your question, the first instance to consult is your supervisor/adviser, and the second one would be your department/faculty administration.