Yes, you're pretty spot-on. "Cinema" refers to visual art captured on film. It never refers to plays or live performances, nor plays of performances caught on film unless specifically adapted to have qualities of cinema. "Theatre" refers to a greater many things, chiefly being live plays and performances, while "a theater" defines the building that houses such performances. "Theatre" is also a discipline of study whose definition varies from school to school, but sharing in common the feature of acting.