Are cell biology and cytology the same science?
Your professors are probably using cytology to refer to cytopathology which isn't strictly incorrect, but is a bit confusing since cytology also has a broader meaning.

Overall, the distinction isn't scientifically that important. Biology is so interconnected that you can't really stick strictly within any one "type" of science, they are only soft distinctions. If you need to learn a term for a course, there isn't much point to argue with your professors. Literally, the word "cytology" should refer to the study of cells, so in my opinion the narrow definition doesn't make sense, but like other words it is important to interpret it in context.

In my personal experience, I have usually heard "cell biology" used rather than "cytology" to broadly mean study of cells. If you are at an institution that calls their cytopathology labs "cytology" it doesn't much matter if there are other meanings for the word.