或: how did "region" become "either"? The character 或 is used today for the meanings "or", "either", "maybe", and "perhaps". Historically, it had the meaning of "region", and it is this meaning that characters like 國 and 域 derive from.

This original definition is reasonable given its composition; it has the dagger-axe radical (戈), and according to 說文解字, it signifies someone guarding territory:


  从囗从戈，以守一。一，地也。


Therefore, it's unclear how this character could get its modern meaning. How did this evolution take place?
without implying that the classical explanation is necessarily correct, worth noting that 康熙字典 thinks 或 was the original single character for 惑, before the latter character was introduced. Not hard to see how the grammatical usages of 或 could have evolved while it still also meant 惑. Or the other way around, the grammatical function could have been original, from which the meaning of 惑 is easily derived.