I think you are a bit confused by different terminology used in SharePoint around permissions.
Let me try to clarify the things.
SharePoint has a notion of
permission - it's a minimal piece of right the user might have. For example user can edit list item, delete items, manage list, etc. Every piece of "right" represented as one single permission:
- Add Items
- Edit Items
- Delete Items
- Manage Lists
- etc. the list is long
In order to give a user set of permissions, instead of assign each one-by-one, there is a notion of permission level. Permission level combine multiple permissions into one single unit. This permission unit can be later assigned to a user (or group) at the specific level (web, list, item).
For example there is a permission level "Read", it includes following permissions - View Items, Open Items, View Versions, Create Alerts, View Pages, etc. By giving user "Read" permission you are giving whole list of permissions inside "Read" permission level. You can give this permissions to user directly, or through the group (when a user is in the group and group has above permission level).
Here is the list of default permission levels.
Now site collection administrators. There is no "Site collection administrator" permission. Think about site collection admin as a privilege. This privilege gives you "Full control" permission level implicitly. Along this, it gives you some additional administrative options, not available for "Full control" permission level.
The page for checking permissions simply showing permission, and since there is no "Site collection administrator" permission, you will not see it. Technically, in term of permissions, site owner and site collection administrator are equal. But as said before, site admin has some advanced administrative options. When you didn't assign direct access to the site admin, you will see the message that "The following factors also affect the level...." on the check permissions page, because site collection admin is always has Full Control permission level, regardless added directly or not.