"Tyrant", or rather tyrannos, was a Greek word for any absolute ruler whose power was obtained by gift, usurpation, or popular revolution. That is to say, the difference between a king (basileus) and a tyrant is that a king's power is inherited; a tyrant's power is obtained. Otherwise, both words refer to an autocrat. Greek tyrants did also bequeath their rule to their descendants, but the descendants were generally called "tyrant" too because the title had become a tradition by that point. Most Greek tyrannies didn't last more than a couple of generations anyway; Syracuse was an exception in that respect.

Syracuse and its tyrants were certainly Greek, because like most other major settlements in Sicily and coastal southern Italy it was a Greek colony.