Which is the correct version of Jason and the Argonauts? The story where Aeson is driven from the throne by his brother, or the one in which he tries to reclaim the throne from his uncle, who stole it from his dad?

This is really a major difference between mythology or ancient epic and the literary works to which most students are accustomed. In written literary works or sacred texts, there is a single "authoritative" version and scholars work hard to establish it. This is normally held to reflect either the "author's final intention" or the readings found in the best "copy-text." Mythology, however, works differently.
In mythology and oral epic, stories were recited. They were improvised by bards in performance. Although the story of Jason (or Achilles or Athena) normally has several common features in all its instantiations (as individual performances are technically called), bards may create or vary details depending on local traditions, audience preferences, or their own artistic visions. Thus while we have more or less common elements in a story, none of the variations are actually "correct" or "incorrect"; they are just different.
In the case of the story of Jason, the main characters are mentioned as far back as Hesiod and Pindar. The two most important literary works that elucidate the story are Euripides's Medea and the Argonautica of Apollonius of Rhodes. In both of these, Aeson, father of Jason, was deposed and imprisoned by his half-brother Pelias.