“Train” usually refers to that equipment that is compatible with the larger national mainline network, which is regulated nationally so that signaling, size, and safety are consistent (e.g. two trains from different companies crashing into each other is not a disaster, they fit on each others’ tracks and stations, they can talk together on the same radio frequency and hook into the same cab signals, etc.)
Metros tend to be isolated systems doing their own thing and are not easily compatible. In the most extreme case like London Underground, they are wildly different dimensions than mainline trains.
Tokyo does have lines like these in its subway umbrella; the Ginza Line and Oedo Line cannot host through operations. It’s not common though, Japan is at the forefront of blending the two, which is often desirable for intraurban operations.
> In the most extreme case like London Underground, they are wildly different dimensions than mainline trains.
Though on the other hand especially in the case of the London Underground the boundaries are actually quite blurry, too.
First of all it's only the deep tubes that are running those noticeable smaller trains, and even then the Bakerloo line nevertheless actually has some through-running onto the mainline railway network, too (Queen's Park – Harrow & Wealdstone).
Secondly, the subsurface lines have some more through-running (District Line to Wimbledon and Richmond and most noticeable the Metropolitan Line north of Harrow-on-the-Hill) and historically, the boundary between the Underground lines and the mainline network used to be even fuzzier, e.g. for a time some District Line trains used to run all the way to Southend and Shoeburyness, beyond today's Circle Line the so-called Middle and Outer Circle services partially ran on mainline railway infrastructure in West London, etc. etc.
Even as late as 1988 there was a special railtour on the subsurface lines utilising standard Gatwick Express Mk2 railway carriages (albeit hauled by London Underground battery locomotives).
Ultimately rail isn’t really a bunch of distinct categories but really a multiplane continuum of various features that people choose to implement or not implement based on local context, desirability and feasibility.
“Train” usually refers to that equipment that is compatible with the larger national mainline network, which is regulated nationally so that signaling, size, and safety are consistent (e.g. two trains from different companies crashing into each other is not a disaster, they fit on each others’ tracks and stations, they can talk together on the same radio frequency and hook into the same cab signals, etc.)
Metros tend to be isolated systems doing their own thing and are not easily compatible. In the most extreme case like London Underground, they are wildly different dimensions than mainline trains.
Tokyo does have lines like these in its subway umbrella; the Ginza Line and Oedo Line cannot host through operations. It’s not common though, Japan is at the forefront of blending the two, which is often desirable for intraurban operations.