There are two Publishing features that need to be enabled, one at the site collection level and one at the web (aka site) level.
A key concept to understand here is that when you create a site collection, it has what is known as a "Root Web", which is the top site in the site collection. Because it's the top site in the site collection, it has some special things which subsites don't have, but, at the same time, it has some things in common with subsites.
Do not confuse the "Root Web" of a site collection with a "root site" of a web application.
In your case, if your site collection has the URL
then that is also the URL of the Root Web of your site collection.
The site at
is known as a "root site" also because it is at the root of your web application.
So, getting back to your site collection, if you go to
and go to Site Settings -> Site Collection Administration -> Site Collection Features, you can enable a feature called "SharePoint Server Publishing Infrastructure". This is the feature you need to enable at the site collection level (hence "Site Collection Features").
Now, if you go back to the Site Settings page, under Site Actions you will see "Manage site features". If you click on that link, you will see a completely different set of features. The one you want to activate there is "SharePoint Server Publishing". That feature is the one that enables Publishing for the Root Web of your Site Collection.