I do not think a layman would think of a train network upon hearing MRT except in Bangkok, Singapore, and other cities where it happens to be part of the official name.
As you can see from Wikipedia's page on passenger rail terminology, there is no single common word for these transportation systems (beyond train or railway perhaps), because there is no universally agreed upon set of characteristics for them, because local usage varies from the definitions of transportation planners, and because individual systems themselves blur the lines and contradict parts of their own names. If you do choose a common name, you will need to define it.
Local usage prevails. In Chicago, the subway refers to the underground portions of the Blue and Red lines of the "L." On the other hand, many Philadelphia natives refer to the whole Blue Line, whose downtown portion is entirely underground, as the El. In New York, the entire urban system operated by the MTA is called the Subway, including both underground and elevated portions. Yet no part of the coexisting PATH system, underground or above ground, is called the subway; it is simply the PATH (or for an older generation, the tubes). Confusing the PATH for the Subway just because it is a subway will mark you as a tourist. Washington's Metro is a regional, not a metropolitan system, and the Metropolitan Atlanta Rapid Transit Authority operates many more non-rapid (i.e. bus) routes than rapid (i.e. rapid rail) ones.
You'll also encounter problems of overlapping and conflicting terminology in discussing freeways / expressways / motorways, but both of these pale in comparison to the debates over what constitutes a car / motorcar , automobile, or motor vehicle.