I don't know that there is a single definition. Non profit doesn't work - I don't think there are a lot of theaters - professional or amateur - that make money.

The quality of the production does not work at all - some of the best productions I have seen have been in community theaters.

Neither does the level of committment - both theaters are kept going by people who love the theater beyond all reason.

Same for "professional" or "amateur" actors and directors works either - there are not many actors anywhere who make a living at it.

I think the defining element is community support - volunteers doing everything from acting and directing to cleaning toilets before a show. A very few people in community theater get paid (besides paid staff) yet people donate time and money to keep things together.

I also think that community theaters are located in the smaller communities and are a matter of community pride. I travel a lot and I visit small theaters (amateur and professional) all over the country. Most of these small theaters are located in very small towns - some can hardly be called theaters - yet they have a loyal group of members or supporters.