The train is merely an intermediate capital good used to provide train seats. It's private at the level of capital goods which is also the level at which it's paid for - only one railway can use a piece of rolling stock (rivalrous) and that railway can prevent other railways from using it (excludible). You might as well say that a house or a car is a public good, since they are merely capital goods that provide a stream of an underlying consumer good (space to live, road trips).
What are the positive externalities of a train? If anything, it's externalities seem to be negative - it creates pollution and noise. (It might be less negative than driving but that's a different claim.)
Private property law allows owners to enforce their own rules(to a point). This both explains the private control over the train and tracks and lends weight to the possibility of being kicked off the train, as it is justified in terms of disruption to service and resulting monetary losses. To wit, a positive externality of transit service is economic gain from the population having improved market access(more goods, services, workplaces within reach). The transit company does not realize these gains, the public does. The company's enforcement of rules is guided by the premise that some amount of security and order is necessary to achieve its service goals.
To wit, a positive externality of transit service is economic gain from the population having improved market access(more goods, services, workplaces within reach).
This is a positive externality of any transit method, including privately owned cars.
Ultimately what you've stumbled across is the fact that nearly any good or service will create gains from trade that are not captured by the producer. E.g., video games, cakes and pies, pornographic videos, and kindles all make people happier and nicer to those around them. This is why encouraging economic activity is generally considered beneficial.
Does this make cakes, pies, video games, pornos and kindles public goods?