Remove portion of header code from home page only

You can combine multiple conditional statements:

<?php 
if ( function_exists( 'show_media_header' ) 
    and ! is_home() // list of newest blog posts
    and ! is_front_page() // front page, might be a static page
)
{ 
    show_media_header(); 
} 
?>

You don’t need a separate file for that.