Hi Markus,
If you like to display blog image medium on Home page then create a blog page and set to to blog image medium template.
Go to dashboard -> setting -> reading and choose static page. Select blog page from the dropdown and leave blank to post page.
By default post page is Blog image large. If you like to display post page in Blog image medium then code customization is required. In line no 140 you will find below code
$image .= get_the_post_thumbnail( $post->ID, 'featured', array( 'title' => esc_attr( $title_attribute ), 'alt' => esc_attr( $title_attribute ) ) ).'</a>';
replace it with
$image .= get_the_post_thumbnail( $post->ID, 'featured-medium', array( 'title' => esc_attr( $title_attribute ), 'alt' => esc_attr( $title_attribute ) ) ).'</a>';
We recommended not to change the code because if you change the code then while updating to our new version all your customisation code will be lost. So better make child theme and first unhook the functions and the edit the code.
http://codex.wordpress.org/Child_Themes
Thank you