Custom Login Errors and variables I can use

There may be an easier way to do this, but you could rewrite the wp_authenticate_username_password() function in the \wp-includes\class-wp-error.php file.

Of course you cannot edit the function directly, but since this function is a filter function, you can remove the filter, copy it, add the copied function back in and test it. After testing that it still produces the same error, you can then change the error in the copied code.

The following code removes the old WordPress function and creates a new function with a unique name.

remove_filter( 'authenticate', 'wp_authenticate_username_password' );
add_filter( 'authenticate', 'wpse_115539_authenticate_username_password', 20, 3 );
/**
 * Remove WordPress filer and write our own with changed error text.
 */
function wpse_115539_authenticate_username_password( $user, $username, $password ) {
    if ( is_a($user, 'WP_User') )
        return $user;

    if ( empty( $username ) || empty( $password ) ) {
        if ( is_wp_error( $user ) )
            return $user;

        $error = new WP_Error();

        if ( empty( $username ) )
            $error->add( 'empty_username', __('<strong>ERROR</strong>: The username field is empty.' ) );

        if ( empty( $password ) )
            $error->add( 'empty_password', __( '<strong>ERROR</strong>: The password field is empty.' ) );

        return $error;
    }

    $user = get_user_by( 'login', $username );

    if ( !$user )
        return new WP_Error( 'invalid_username', sprintf( __( '<strong>ERROR</strong>: Invalid username. <a href="https://wordpress.stackexchange.com/questions/115539/%s" title="Password Lost and Found">Lost your password</a>?' ), wp_lostpassword_url() ) );

    $user = apply_filters( 'wp_authenticate_user', $user, $password );
    if ( is_wp_error( $user ) )
        return $user;

    if ( ! wp_check_password( $password, $user->user_pass, $user->ID ) )
        return new WP_Error( 'incorrect_password', sprintf( __( '<strong>ERROR</strong>: The password you entered for the username <strong>%1$s</strong> is incorrect. <a href="https://wordpress.stackexchange.com/questions/115539/%2$s" title="Password Lost and Found">Lost your password</a>?' ),
        $username, wp_lostpassword_url() ) );

    return $user;
}

The next step is to test the code to make sure it is producing the same error as before. Once tested, try changing the error message as you wish. If things go too wrong, start over or comment out the first 2 lines and the WordPress function will use the old error message.

// remove_filter( 'authenticate', 'wp_authenticate_username_password', 20, 3 );
// add_filter( 'authenticate', 'wpse_115539_authenticate_username_password', 20, 3 );