@Edward
Hope you made a backup (sql dump) of your database that was used with the 2.0.+ you had. Reason ... if everything worked as is should, one should have seen an upgrade process ... not an installation process. Upgrade processes don't show the license screen again ... it prompts you that you are about to upgrade from X to Y and do you want to continue.
From what you've described (without screen snaps) sounds like the installation started, but failed to finish. Uhhh, bad news if it has already updated some tables in the db but not all. Too much for one human to do via phpmyadmin, for sure.
To be honest, check the download.moodle.org page again for the version you downloaded:
"This is the last formal release of the 2.0 branch. Support for this branch has been discontinued. We highly recommend you upgrade!". That version will not be getting any security updates either. You might point that out to your hosting provider. PHP requirements aside, they are not providing their customers the level of service needed in regards to security of apps they install.
You might be better off all the way round shopping for another provider for your online Moodle. One that will install for you (if that's what you need) a more secure version.
'spirit of sharing', Ken