When I say "train station" I am corrected to say "railway station", but I don't understand why.
Buses are stationed at a bus station.
Trains are stationed at a train station.
Why in this respect is "train station" incorrect?
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In short, don't worry about the difference, unless you're trying to sound old fashioned (e.g. writing historical fiction). In general, "train station" is fine.
However, if you're trying hard to please some particular person who is correcting you, it may be helpful to use the particular term they prefer, using the word that is most effective at conveying to that particular listener what you are trying to convey. (See relevant XKCD here.)
This is a question of usage.
in British English, up until a couple of years ago it would always have been Railway Station.
However, in recent years, Train Station has entered popular usage. Whether this is borrowed from American English, I am not able to say.
However, to address your question, both are correct. It is probable that your teacher seeks to preserve the Railway usage as that is what he is used to.
Although they essentially mean the same thing, here in the United States (Boston) there's a subtle difference. We have four different kinds of trains: intercity (Amtrak), suburban commuter rail, urban "rapid transit" subways, and urban light rail.
The term "train station" could be used for all kinds of trains, but "railroad station" (or railway station) would typically be used only for the longer distance intercity and commuter rail trains. Train stations for the subway and light rail are often called "subway stations", even for trains that run above ground. Locally, they're more often called "T Stations" or "T Stops". ("The T" is our local nickname for the transit system operated by the MBTA transit agency.)
A century ago, there were many different railroads that competed with one another. Although there were some shared "union" stations, the railroads often built their own stations for exclusive use by their own trains. For example, in New York City, Penn Station was originally built by the Pennsylvania Railroad and Grand Central Station was built by the New York Central Railroad.
Because these grand and fabulous stations were built by and belonged to the railroad companies, it may have seemed more appropriate to call them railroad stations (or railway stations), instead of just train stations. Nowadays, there are no great passenger railway companies; it's just various agencies or other entities that operate trains. So, that may be why "train station" is now the most commonly used option.
In most parts of Britain until around the late 1980s people wouldn't say 'train station' or 'railway station'. When referring to railways they'd just say 'station'. The word would only be qualified for bus station, fire station, tube station, etc.
Station was synonymous with railways, many towns and cities have a Station Road or a Station Hotel, not a Train Station Road or Train Station Hotel.
In Britain and Australia, it has always been "Railway Station" and not "Train Station" until the last 20 years or so. Usually, trains only stop momentarily, to pick up or set down passengers but the are NOT stationed there. Hotels adjacent to Railway Stations are usually called either the "Railway Hotel" or the "Station Hotel" but I have not come across a "Train Hotel". I think the gradual change to "Train Station" is either laziness or people who watch a lot of American Movies, picking up the American usage.