"I then set up a User and Password identical on the remote host."
If you began ny setting up the database on your local computer, I don't understand how you can then set up a user and password 'identical' on your remote server?
Did you use your remote hosting username and password when you installed mysql on your machine? I use IIS - so I'm unfamiliar with how other server models are configured for testing purposes.
It is often the case that your mysql login and username is partially set from (prefixed by) your Hosting account username.
so my local connection is
$hostname_mymachine = "localhost";
$database_mymachine = "cms";
$username_mymachine = "root";
$password_mymachine = "XXXXXXXX";
whereas my hosted mysql is
$hostname_localhost = "www.pictuality.co.uk"; (or localhost)
$database_localhost = "uks6XXX6cms";
$username_localhost = "uks6XXX6nh";
$password_localhost = "XXXXXXXX";
[uks6XXX6 being my host-assigned prefix]