If one event lies outside the light cone of another, can the events to some observers appear in a different order in one reference frame compared to the other, and is this the only time when this is the case? (I think this is right but am looking for verification)
Events which lie within each other's light cones are called "timelike separated." All observers agree on the ordering of these events.
Events which lie on each other's light cones are separated by "lightlike" or "null" interval. All observers also agree about the time ordering of these events.
Events which lie outside of each other's light cones are called "spacelike separated." Different observers may disagree on the order of these events.
So yes, you are correct. Only events separated by spacelike intervals may be re-ordered by changing reference frames.