Trying to understand IIS permissions
Solution 1:
The group BUILTIN\Users has Write Access to your C:\Site1-Logs directory.
The user IIS AppPool\Site1 is automatically a member of the Users group because that is a special group under Windows.
This is why your web-site has write access.
You can't remove IIS AppPool\Site1 from users, but you could remove the permissions of the users groups from the directory.