Further, in the big metro areas, the private trains do just fine.
JR East is #1, Tokyo Metro is #2, JR West is #3, Tokyu is #4, ... the next JR, JR Central is down at #9 with #5 #6 #7 #8 all private. Tokyo Metro is private, Toei (is the city run subway, it has 4 lines as is far down the list).
https://en.wikipedia.org/wiki/List_of_through_trains_in_Japa...
Asakusa (one of "my" lines!) line, is definitely a subway inside central Tokyo, but you can stay on the same physical train going all the way from Narita to Haneda (think RER in Paris?) — I think it would be qualified as "light rail" anywhere else in the world.