When the Amphitheatres were being built, the designs were all about the projection of sound ands how it was Auditorius for those in the seating provided.
Auditorius become in modern terminology a Auditorium is a word with a few hundred years origin, which is a bit of a Hybrid term covering both Amphitheatre and Auditorium.
Audio as a word is much more modern, relating to electronics involved in producing sound.
It looks like sound has a very long history of being produced for an assembly of individuals to receive, maybe I am a traditionalist in my approach, and lucky to know quite a few like minded traditionalists.