[WordPress] How to correctly filter wp_title for a customized format.

correct-wp-title.php

<title><?php wp_title( '|', true, 'right' ); ?></title>

filter-wp-title.php

<?php

/**
 * Provides a standard format for the page title depending on the view. This is
 * filtered so that plugins can provide alternative title formats.
 *
 * @param       string    $title    Default title text for current view.
 * @param       string    $sep      Optional separator.
 * @return      string              The filtered title.
 * @package     mayer
 * @subpackage  includes
 * @version     1.0.0
 * @since       1.0.0
 */
function mayer_wp_title( $title, $sep ) {
	global $paged, $page;

	if ( is_feed() ) {
		return $title;
	} // end if

	// Add the site name.
	$title .= get_bloginfo( 'name' );

	// Add the site description for the home/front page.
	$site_description = get_bloginfo( 'description', 'display' );
	if ( $site_description && ( is_home() || is_front_page() ) ) {
		$title = "$title $sep $site_description";
	} // end if

	// Add a page number if necessary.
	if ( $paged >= 2 || $page >= 2 ) {
		$title = sprintf( __( 'Page %s', 'mayer' ), max( $paged, $page ) ) . " $sep $title";
	} // end if

	return $title;

} // end mayer_wp_title
add_filter( 'wp_title', 'mayer_wp_title', 10, 2 );

wp-title-incorrect.php

<title><?php bloginfo('name'); ?> | <?php is_home() ? bloginfo('description') : wp_title( '' ); ?></title>