They are roughly the same, the differences are very vague if any.
Мятеж is more focused on the goal (overthrowing the authority) and as such is used in legalese, бунт is more about the process itself. The difference is roughly the same as that between "rebellion" and "riot".
Star Wars' rebels are мятежники (or повстанцы), "Rebel without a Cause" is Бунтарь без цели.
Бунтарь, as your dictionary mentions, can be used figuratively:
Можно быть великим бунтарем для всего мира, как Ибсен, а жить в мещанской обстановке, так что никто из ближайших соседей и не узнает, что жил тут великий бунтарь. // М. М. Пришвин. Дневники (1918)
, бунтовщик would be less appropriate here (though still could be used).