Learn English – The correct usage of very vs a very

According to Randolph Quirk et al.'s book on English grammar, structures like X is Y, where X is a noun describing the noun X are in "a copular relation". Y here is called a complement.

We put an indefinite article before complements in such structures if Y is not unique: "Mark is a programmer". However, if the complement names a unique role or task, we may use either a definite or a zero article: "The user is (the) administrator" (that would imply that there's only one administrator position); "John is (the) captain of the team."

P.S. Y may be a complement or an appositive noun phrase ("John, (the) captain of the team.").

A better statement of the definite article "rule" would be that you should use the X only when there is a reasonable expectation that your readers or hearers will understand what particular X is meant.

This includes situations when you have previously mentioned X; but it is not necessary to have mentioned a specific train, because few people hang around a railroad station with the plan of taking "a" train, any train that shows up. They're there to catch (or meet) a particular train. Even on a subway system where trains run on any given line every fifteen or twenty minutes, people are waiting for the next train.

Consequently, the hearer or reader may be expected to infer from context that what is meant is “the” train the speaker/writer is waiting for: the train to or from a specific place scheduled to arrive here at this specific time.