劇場 and 舞台: a theater, a stage, and a setting for activities
Distinguish 劇場 as a theater facility from 舞台 as an actor's stage or a city's figurative role as a setting, keeping door opening separate from watching a play.
劇場 names the theater facility
「駅前の劇場で新しい芝居を見た。」 says that a new play was seen at the theater near the station. 劇場 names the facility where the play was watched, and で marks that location.
「劇場の扉は開演の三十分前に開く。」 says that the theater doors open thirty minutes before the performance. 開く describes the doors opening, without reporting that the performance has been watched. The timing belongs to this example, rather than an operating rule for all theaters.
舞台 names an actual stage or a figurative setting
「俳優が舞台に上がると客席が静まった。」 says that the audience fell silent when the actor stepped onto the stage. 俳優 is the person moving onto it, and 客席 is the audience side that became quiet. 舞台 names the actual stage.
「この町は新しい活動の舞台として注目された。」 says that this city attracted attention as a setting for new activities. 舞台 figuratively names the place where activities can unfold. It does not report construction of a new physical stage in the city.
Separate the facility from the stage's two meanings
劇場 is the theater facility where the new play was seen. 舞台 can be the stage the actor stepped onto or the city's figurative role as a setting for new activities. Their connection with performance does not make the facility, physical stage, and figurative setting identical objects.
In 俳優が舞台に上がる, follow the actor's movement; in 町が活動の舞台, follow the city's role. Door-opening time gives information about the theater, rather than identifying who stepped onstage or confirming that a play was watched.
What does 舞台 refer to in 俳優が舞台に上がる and 町が活動の舞台, and how is it different from 劇場?
Show the answer and why
The first 舞台 is the actual stage the actor steps onto; the second figuratively presents the city as a setting for new activities. 劇場 names the theater facility where a play was seen. The city's figurative role adds no newly constructed stage, and opening the theater doors does not confirm completion of watching a play.