Let's stick to the definitions from The British Theatre Guide and then part from there.
According to TBTG, drama is a branch from theater based on a representation of an event written in prose or pantomime that is meant to duly enact and illustrate an event or situational conflict through acting.
A play is the conduit upon which the event is represented. It consists on stage directions, dialogue, prologue or preface, author's notes and even the dramatis personae who will be included in the drama. Drama would be the opposite of Comedy, which is also a lifelike representation that views conflict under a lighter, jovial and paradoxical light.
The theater is the scenario or the stage on which the play drives the drama. The drama contained in the play unveils and expands on stage. Notice that even in the military the term "theater" is widely used as
An area defined by the geographical commander or Department of Defense as the place where a specific operation will take place.
Circa 1660, the meaning for theater was "place of action" as well as "region of war" therefore, special attention should be given to the meaning of "theater" as an actual place or location, and not just as the meaning of the art itself. "Theatrical Arts" would be the correct usage for the word "theater" as a branch of humanities.