Warning: file_exists(): open_basedir restriction in effect. File(/home/vhosts/moodledata) is not within the allowed path(s): (/home/vhosts/EXAMPLE.COM/:/tmp/:/usr/share/
pear/) in /home/vhosts/EXAMPLE.COM/install.php on line 329
Warning: is_writable(): open_basedir restriction in effect. File(/home/vhosts) is not within the allowed path(s): (/home/vhosts/EXAMPLE.COM/:/tmp/:/usr/share/pear/) in /home/vhosts/EXAMPLE.COM/install.php on line 333
say it all. a) the web server has no rights to create moodledata, which you can help by manually creating it. b) open_basedir of the web server is in effect which generally gives problems with Moodle.
P.S. Your screen-shots are very clear, perhaps too clear. I don't think exposing the true
URL of a broken site is a good idea.