Guys, thank you for all the input!! IT'S WORKING!!!
The solution was to edit the vesta-templates of basedir.
1. First we copied the moodledata to /home/[myusername]/web/[mydomainname]/moodledata
2. Then we changed the path in the config.php in the main moodle folder
3. Then we changed the basedir.tpl and basedir.stpl templates in /usr/local/vesta/data/templates/web/apache2
ORIGNAL:
php_admin_value open_basedir %docroot%:%home%/%user%/tmp
EDITED:
php_admin_value open_basedir %docroot%:%home%/%user%/tmp:%home%/%user%/web/%domain%/moodledata/
Now everything is fine. If you have the same problem be sure that you select the right template for the user in VESTA, in my case it is the "basedir" template. You canb also copy those files to create a new template.
Thanks a lot to all little helpers and the funny pictures!! :D