You want to edit the title, and not the content.

Check this for filters you want to use.

If you want to edit your title you should add_filter to it and modify it(the_title). Like so:

add_filter('the_title', 'addBadge2Title');
function addBadge2Title($title)
{
    $seconds = strtotime("now") - strtotime(get_the_date("Y/m/d"));
    $badge = get_stylesheet_directory_uri() . '/library/images/new_ribbon.gif';
    if ($seconds < 10950400) {
        $title="<img class="new_ribbon" width="75"  height="75" src="" . $badge . '" >' . $title;
    }
    return $title;
}

Edit: use badges only for specific categories

If you want to use this code for specific categories you have a few options

1. First remove add_filter('the_title', 'addBadge2Title'); and keep only the function declaration in functiosn.php and in the archive-{$category-slug}.php file add and remove the filter like so

   if(have_posts()) : while(have_posts()) : the_post()
   
       add_filter('the_title', 'addBadge2Title');
       the_title();
       remove_filter('the_title', 'addBadge2Title');
   
   endwhile; endif;
   

2. Check if the current post has the category you want the title to be displayed with badges. Modify the function by adding has_category like so:

   function addBadge2Title($title)
   {
       // return the $title unmodified if the post does not have the category
       if(!has_category('badge-category'))
           return $title;
   
       // add the badge
       $seconds = strtotime("now") - strtotime(get_the_date("Y/m/d"));
       $badge = get_stylesheet_directory_uri() . '/library/images/new_ribbon.gif';
       if ($seconds < 10950400) {
           $title="<img class="new_ribbon" width="75"  height="75" src="" . $badge . '" >' . $title;
       }
       return $title;
   }