You should be able to get the categories_description field in $categories by adding it to the original query in /includes/modules/pages/index/main_template_vars.php (no override path available).
Add , cd.categories_description to the query in two places, line 96 and line 110:
Code:
$categories_query = "SELECT c.categories_id, cd.categories_name, c.categories_image, c.parent_id, cd.categories_description
You should now be able to use $categories->fields['categories_description']
Note that the previous standalone query in posts 7 & 16 would not work if your database has table prefixes, or you use multiple languages. The query in main_template_vars accounts for those cases.