Theatre and amphitheatre (or theater and amphitheater) have the distinction that a theatre is one-sided viewing of a central stage, and an amphitheatre has viewing all around. The word amphitheatre derives from the Greek word amphi- loosely meaning on both sides or all around.
For years, I've used amphitheatre for any large open galleried seating, but technically, as for the spectacular Minack theatre in Cornwall, England, if it's one-sided, then it's a theatre.
You'll also see the same pattern in amphoras, which commonly have handles on both sides of the vessel, and amphibians who are at home on both land and water. Plus, related ambivalence, ambidextrous (amphi- Greek, ambi- Latin).
It's fun having an aunt who used to teach classics =)