Support

You have locked yourself out of your site. First you need to follow these instructions to unblock yourself: https://www.akeebabackup.com/documentation/troubleshooter/atwafissues.html

Then you can use the correct administrator secret URL parameter.

We recommend only using lowercase and uppercase letters a-Z and numbers 0-9 in the secret URL parameter. If you use special characters they need to be URL-encoded when you try to access the administrator page. Due to the difficulty of doing that manually and compatibility issues with browsers and servers you are likely to have issues if you don't use just a-Z, 0-9.

OK, these are two completely different things.

File ownership: Admin Tools has nothing to do with it. That's something configured in php.ini by your host OR that's your Linux user ID (if you have enabled Joomla's FTP layer). Please ask them to take a look into it.

Administrator secret URL parameter: Try setting it to test123 (all lowercase). Then access your site as http://www.example.com/administrator/index.php?test123 where http://www.example.com is the URL to your site. Kindly note that administrator is followed by /index.php. According to my experience a host which uses the sort of mixed ownership you described has also set up Apache in such a way that the /index.php bit is required.

If that works try using a secret URL parameter of your liking. Please remember that it's case sensitive. This means that abc, ABC, Abc and aBc are four different secret URL parameters. That's inherent to the way your web server and PHP itself works. The secret URL parameter is, in fact, the same as the any other parameter. For example, index.php?option=com_login has one parameter called option with the value com_login. If you pass index.php?option then you have one parameter called option with an empty (null) value. The secret URL parameter feature works by checking the existence of the parameter whose name you've entered and make sure that it's value is empty. It's therefore unadvisable using any system-important names like option, format, tmpl, tpl, tp, view or task. These have special meaning for Joomla and are automatically populated with non-empty values. Therefore using any of these will break your login. To be honest that'd be the first time in 8 years someone has tried to do that but one thing I've learned in my 15+ years of web development experience is to never consider anything impossible, however unlikely :)