Is the word "coin" a proper bitcoin term? I have noticed that some people speak of "coins".  As far as I can tell, by "coin" they mean a transaction output.  Is this a correct understanding?  Is it helpful to use the word "coin"?
Right.  The statement "Alice has 5 coins" could be more precisely phrased "Alice controls the private keys for some set of addresses, to which are directed some set of transaction outputs with a total value of 5".  But that's a lot more cumbersome.