Hi,
We've finally managed to get it working thanks to your advice with repairing boot files. I went to create the Windows 8.1 startup disk, go to its command line, enter the bootrec commands, destroying the backup system booting and then try to update everything via the Windows Update. In the end all remaining 8 important and 4 optional updates have been installed. Just when entering bootrec commands, the /fixboot and /rebuildbcd parameters resulted in the "element not found" message while the other two, /fixmbr and /scanos worked but the next boot was a real failure resulting in the "insert boot disk" message. We then had to boot the startup disk again and let Windows to repair booting automatically which worked and allowed Windows to start again, then it was possible to update everything. Finally I took the installation of the backup system and managed to return it back so the computer is fully operational now. Thanks for your advices and all of your team behind this forum, it really helped and has given me another bits of knowledge.