If a word has two different meanings, is it two different words or one word with different meanings?

This is a question that is discussed by eminent British linguist David Crystal in his article How many words?, in which he attempts to come up with a reasonably accurate count of the number of words in the English language. Basically, he says that no accurate number is possible because there is no universally-agreed definition of what constitutes a word. Here is an extract that addresses your question, but provides no definitive answer:

Is the lock on a door the same basic meaning as the lock on a canal? Should ring (the shape) be kept separate from ring (the sound)? Are such cases 'the same word with different meanings' or 'different words'? These are the daily decisions that any word-counter (or dictionary compiler) must make.

The article is here (pdf): www.davidcrystal.com/?fileid=-4890

Both possibilities are can be found. A word can get a new meaning simply by semantic change. That is, a word can take on meanings that are closely related as for instance the consequence of a thing or an act. There are a lot of possibilities how a word can develop new meanings.

The second possibility is two words have become identical in the course of time due to historical sound change or drop of syllables and other reasons. An example is the adjective fresh in the sense "fresh from the press/the oven". Most dictionaries have the meaning "insolent" in the entry of fresh. But that is actually a second word and it would need a second entry. "fresh" number 2, mostly in AmE, as in "Don't get fresh" must have been invented by German-American speakers who anglisized German frech (insolent) as "fresh". It is not seldom that one finds similar things, I mean that in a dictionary entry two different words are contained.

Actually, I didn't answer your question exactly, Laura. You asked whether train (noun, on railroads) and the verb to train someone (to teach) are the same word. The question is reasonable. But as in English dictionaries noun and verb and other word classes are treated in one entry one should say it is one word, especially as the etymological source is the same: Latin trah-ere to pull and according to etymonline from Vulgar Latin *traginare.

A locomotive pulls the wagons/carriages a train consists of. A coach training his team pulls them towards an efficient state of achievement.

But my personal view is that a noun is one word class and a verb another word class. So I would prefer the formuluation "a train" and "to train" are two different word classes from the same historical source.