From the wiki, "a coin is a number associated with an address." Fine, but this doesn't answer my question. What form does the record by which you verify that an individual who has spent a coin no longer owns it take? Is it simply the latest associated address?
If that coin is one of the inputs to a later valid transaction, then it is spent and cannot be spent again.
That later transaction is the record that somebody spent the coin; it is signed with the private key, and that key (which Bitcoin keeps for you in your wallet) should be known only to the owner.