In this case, 'self-fulfilling prophecy' is just a common phrase with a specific meaning. In general, a prophecy and a prediction are the same. Though we usually use prediction for more normal things and prophecy for very profound or divinely inspired things. The prediction about the rain didn't come true. His prophecy of the great floods didn't come true. There's also the phrase 'prophecy of doom', which is used when people predict very negative things.

A ‘self-fulfilling prophecy’ is a term with its own meaning in popular culture. It says here that pessimism about personal outcomes leads to negative outcomes, and can also mean the reverse - that optimism leads to positive outcomes. It’s not true, however. Optimism and pessimism correlate with outcomes because there are observable predictors of success which create correctly optimistic and pessimistic beliefs in the first place. Incorrect beliefs have negative effects in that falsely optimistic people waste a lot of time trying to succeed but still fail, whereas the falsely pessimistic give up on something which they would have succeeded at.

You cannot use "prediction" to describe expectations that are mere intuitions.

prophecy usually means some kind of divine telling of the future

prophecy and prediction are similar in the sense they both foretell the results of certain events yet to take place. However the difference is the source. Prophecies come from spiritual gods or religion beings, belief system.e.g bibical prophecies, visions or dreams. Predictions are made based off of scientific data, statistical data or trends of past events. e.g weather predictions, sports match predictions, election results predictions. In this question prophecy is the best answer because the source of the outcome stems from negative thoughts of the pessimists which not based on any factual data but as a result of their own belief system and thoughts.