Thx for all the replies LOL.
I spent a lot of time over the weekend finding the solution to this problem, but I'll only relate the most useful steps - and the solution. Since no-one else could help me, this post is for any Adobe software user who had the same problem.
1. I restarted my Mac in "Safe" mode and I got a different message from the Russian dialog box. The new message box said my Illustrator was suffering from "Error 16" which enabled me to do another search in Discussions and Adobe help. I found that Error 16 has been around since 2014, probably since the dawn of CC.
2. The solution is simple. Reinstall Illustrator CC, open it then sign out. I had uninstalled CC earlier and this did not solve the problem, so I suggest that you uninstall and reinstall it. Illustrator CS6 now works! I checked it again by restarting my Mac and CS6 was still working.
3. There may be other reasons for Error 16 so here are a few urls on the subject. Some of the discussion threads are quite amusing:
https://forums.adobe.com/message/10492120
https://helpx.adobe.com/creative-suite/kb/install-log-error-failed-open.html
https://helpx.adobe.com/x-productkb/global/troubleshoot-system-errors-freezes-mac.html
https://helpx.adobe.com/creative-cloud/kb/configuration-error.html#restore_permissions
https://helpx.adobe.com/creative-suite/kb/cs6-system-requirements.html
https://forums.adobe.com/message/10477104#10477104
https://forums.adobe.com/message/10492120#10492120
*** https://forums.adobe.com/thread/1584746 *** - How do I prevent Creative Cloud from taking over my CS6 perpetual license?
The last url had the solution at post 37. So thx very much poster 08389sdfsgaerghaer
4. Some users may need to reinstall their CS6 software. I could not get these instructions to work, apparently because there is an incompatible compression scheme for one file in CoreServices:
Installing Creative Suite on macOS 10.12 (Sierra)
(I am using High Sierra) so I started up my Mac in Mavericks OS and installed from there.
5. System requirements for CS6
https://helpx.adobe.com/creative-cloud/kb/configuration-error.html
Where to download CS6
https://helpx.adobe.com/download-install/kb/cs6-product-downloads.html