I think it depends on what the particular rail system itself is called. I'd use 'commuter rail' or 'light rail' rather than 'suburban train' for a system that goes from the city center far into the suburbs. 'Suburban train' might suggest that it does not go into the city itself.
For example, the commuter rail system that serves the Boston, Massachusetts suburbs begins in either of two railroad stations near the center of Boston (North Station and South Station) and stops at more than 80 stations southeast, south, and southwest of the city and about 140 station northeast, north, and northwest of the city.
This is different from the subway-and-trolley system in Boston and its near suburbs.