Modern transit comes from two historic lineages: Steam railroads, and horse-cart street railways. To operate in an urban environment, steam railroads underwent a number of modifications. To reduce conflicts with street level traffic, were either elevated or under-grounded. At some point, they were also electrified, typically use a third-rail system. Once that occurred, expansion required the continued use of grade separated, exclusive guideway, so no one touched the third rail and died. In contrast, street railways were electrified as trolleys, using a pantograph. (Cable-cars can be thought of as a 'dead branch' alternative to electrification). The converging modes of electrified heavy rail and street railways were hybridized as the "Inter-urban". Electrified the whole-way, using a pantograph, and running in a mix of at-grade and tunnels. After the second World War, almost all pure street running 'trolley' systems were 'bus-tituted' out of existence, while some inter-urban systems survived. The survivors all had some off-street running-way, viz: RTA Streetcars, San Francisco cable car, MBTA Green Line & Ashmont–Mattapan High Speed Line, SEPTA Subway–Surface Lines: Suburban Trolley Lines & Girard Ave Trolley, RTA Rapid Transit: Blue and Green Lines, Newark Light Rail, Muni Metro. Between ~1930-1972 is sort of a 'Dark Age' for urban rail--almost nothing new is built. Then there is a resurgence of heavy-rail systems to deal with traffic congestion: BART (1972), Washington Metro (1976), MARTA (1979), Baltimore Subway (1983), and Miami-Dade (1984). All run at-grade in the suburbs, and in tunnels in the city center. About 1980, America adopts the Stadtbahn/'City Rail' concept from Germany, and APTA coins it 'Light Rail'. It runs at-grade in the suburbs, and at-grade in the city-center, like the inter-urbans. Being regulated as 'light' rail, it is allowed to operate in mixed-traffic with cars, making it easier/cheaper to build. Over time, the surviving inter-urbans are rebuilt/revitalized, making use of the same vehicles as the new 'Light Rail' systems. Circa 2001, Portland reinvents the 'streetcar', which runs at-grade, in mixed traffic, with smaller vehicles, and making extensive use of single-track.
Now, to get back to what is 'Rapid' transit: Rapid transit is something that has it's own (unshared) guideway. Subways, elevated rail, commuter rail all clearly meet this standard, as do most of the 'Metro' systems of the 1970's heavy-rail revival. But the surviving inter-urbans and new light rail systems are a confusing mix: They have portions of exclusive guideway, so they have rapid transit portions. But LRT means 'Light Rail Transit' rather than 'Light RAPID Transit'. This gets confusion in the context of BRT, which actually means 'Bus RAPID Transit'. BRT gets developed in Latin America as a sort of bus version of a heavy rail system--buses with unshared guideway. But that's another topic. In summary: HeavyRail = Rapid, Streetcar !=Rapid, LRT !=Rapid...but does have sections that could be. (Cable-cars get lumped in with LRT largely on the basis of Cable-car != heavy-rail.)
And finally: Metro!=Heavy-rail, but Metro ⊂ Heavy-rail. Freight, Metro, Subway, Elevated, Commuter Rail ⊂ Heavy-rail.
= Equal to
!= Not equal to
⊂ Is a subset of.
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