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But let's be clear -- this is a pedagogical decision, not a physics rule.But....and here is why we keep this out of the intro course...you now have a mix of frames. The person swinging the bucket is in a non-accelerating frame whereas one can ride with the water in the accelerated frame. The intro student is going to relate to standing there swinging the bucket. Big confusion, I think, to mix the two. Instead, consider the rotor ride, where the student is 'glued' to the wall of the rotating cylinder. This one is better for analyzing from either point of view, but of course in the accelerating frame you have to give up Newton's laws (especially the third) since there is no agent to which you can assign the centrifugal force.
When I teach 1st year students, I do stay away from rotating reference frames. I tell them that WE will not be including pseudo forces on OUR free body diagrams but that if they continue to study physics in college, they will learn that there are many situations where it is easier and more elegant to work with them.
But when a student says: "The water stays in the bucket because it is held up by centrifugal force", I think I have to accept that as a correct answer. "Experts" use these forces to simplify things. Students should also be allowed to do this as well. But again, that is just another pedagogical opinion.