The Media page is the page that is set as the post page in your site's settings:
The page that is set as the post page will use a template file to display the most recent posts and the post page is not editable in Cornerstone so whatever content you will try to add to the page will be disregarded as the page content will only display the post template content that is why you are not seeing the customization you have added.
If you want to have a post page and be able to customize it, you might want to check out The Grid or The Essential Grid plugins. You can use them to display the posts on your site and you can add them in Cornerstone.
Hope this helps.