I just realised I hadn't really posted any solutions to it yet, even though I have the system back up and running. I'll start off with what I did that DID work, and what I think it may be.
1. I made backup of old database, of course. Being not able to enter my own site to create the backup, I had to use the website hosting functions to do this. Even though the site was down due to the FATAL PHP ERROR, the database, was hopefully, still intact.
2. I deleted existing installation of WordPress 2.3.3 – EVERYTHING
3. I uploaded/installed new installation of WordPress 2.5.1 using existing MYSQL database accounts and settings. Did not work – same FATAL PHP ERROR.
4. There was a suggestion of removing “wp_” from the database table prefix in the config file. So I tried this. Then entering the website showed the “new installation” screen and setup. So I followed it and “installed” again. This got the site going back to original defaults with no data. Attempting to login proved difficult as it showed “insufficient privileges” even though it was administrator account.
5. I created a different MySQL database, fresh with no data. Set permissions for existing user to be able to manipulate and edit the database tables. Then all I changed for WordPress configuration was the database name. Same FATAL PHP ERROR.
6. I deleted the non-working installation of WordPress 2.5.1 and tried the same steps for another installation of 2.3.3 – same results.
7. I used website-hosting service “Fantastico” and installed WordPress. It asked me to delete/remove the existing installation (so I did). As it turned out, the version it installed was WordPress 2.5.0
8. I noticed a new MySQL database, so I migrated all my existing data from my backup database to the new one created by Fantastico. There was also a new database user created to be able to edit the tables. EVERYTHING WORKED AGAIN!!!
9. Spent the next few hours/days updating plugins and ensuring the 2.5.0 installation was working with my site. Functionalities were lost and some pages looked a bit funny. My Theme wasn't quite compatible and ready for plugins to be deactivated.
10. Once things were up and running smoothly – upgraded to WordPress 2.5.1 with no hassles.