I would do like you did in your first try:
setcookie("user", "$_COOKIE[user]", time()-3600); //This should unset it, as long as you set the cookies life to an hour...
Remember that the pages usually needs to be refreshed in order for the cookie unset to take effect; so, if you are making a logout script I would do something like this:
setcookie("user", "$_COOKIE[user]", time()-3600);
header("Refresh: 0; URL=http://" . $_SERVER['HTTP_HOST'] . dirname($_SERVER['PHP_SELF']) . "/" . $url); //replace $url with where you want the user to be sent to.
Let me know if this works. 🙂