HI Eva,
several people are running my plugin succesfully on their 2.3 installations - the latest code is in my git repository here:
https://github.com/danmarsden/moodle-plagiarism_turnitin
the code in the master branch is currently the best version to use for Moodle 2.3 but I'll be branching that in git sometime soon into a MOODLE_23_STABLE branch as we might land some of the work my GSOC student has been doing for Moodle 2.4 to integrate plagiarism with the workshop and forum modules.
you should be aware that my plugin is planned for replacement by a new plugin from Turnitin at some point - as per my blog post here:
http://danmarsden.com/blog/2012/03/06/future-of-moodle-turnitin-plugins/
In the meantime you can still use my plugin but you will need to send an e-mail to Turnitin asking them to enable your account to use my integration - see step 2 on the page you linked to above:
http://docs.moodle.org/23/en/Plagiarism_Prevention_Turnitin_Settings#Activation_of_your_Turnitin_Account
The new code in that repository should work fine with the new assignment module in 2.3 but keep an eye on the commits for any updates/bug fixes.