' 'Bye' is a contraction of 'goodbye'. We use many different phrases, depending on the circumstances, when we say 'Bye to a person. I often say 'take care' but if I knew I was going to see the person soon I would say 'see you soon'.
Bye/goodbye is a very neutral comment. It doesn't have any particular emotional content in itself and has no sense of any of 1-3, for me. That comes in associated comments and in unspoken communication.