It's possible to say "We travel by MTR train," although you could also say "We travel on the MTR train."
With "by" + a means of transportation, the article is omitted:
They crossed the ocean by ship. We're going by car. It's much faster to travel by plane.
Once you modify the noun, though, by adding an adjective or an article, or if you make the noun plural, you don't use "by":
Columbus crossed the ocean IN three small ships. We're going out to New Jersey IN a rented car. John drove me IN his car. I'm leaving ON a jet plane. Helen doesn't travel ON planes. She'd rather go ON a bus.
In your example, if you think of the MTR train as one kind of transportation, you could say "by MTR train." However, if you think of MTR as an adjective, then you'd say "on the MTR train." You do need the article or other determiner if you use a singular noun after "on" or "in" to express means of transportation.
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