I can't compare it right now, because I've launched recovery menu and I've selected Restore File System (I've done it to check if File System is OK). Restoring stopped at 60% and there was a message saying that the file system is corrupted and that my HDD has to be formatted

I've tried "System Update" function from Recovery Menu, but selecting it hangs my system (even if there is no USB drive plugged in). The problem is that I don't have any save game backups, so now I'm stuck at this point.
If you are explicitly asked to reformat your drive, it means the file system is corrupted & it is likely that you won't be able to update or anything else...
What CFW are you trying to install via system update? If it's Rebug, it will most likely fail via recovery, it's a problem that exists for a long time. Rebug should be installed from the xmb.
Try with another CFW like Starbucks Habib 4.81 or Ferrox 4.81.
But given your problem, I doubt it will update ok anyway. You can try...
It could be your internal hdd malfunctioning or something like Ccapi messed with your system.... There are various possibilities...
At this stage, you will probably have to reformat your disk or replace it with another hdd. If you do not have any backup, am afraid your game saves might be lost, along with your licenses & the trophies you did not sync with PSN. Unless you have a copy of eid_root_key?
Can you answer this questions to begin with...?
1. First tell us what CFW is currently installed?
2. Did you install ccapi?
3. Do you have the eid_root_key for your hdd? You should always keep a backup of this key made with Rebug Toolbox otherwise you cannot read the hdd on a pc. If you ever converted cex/dex with the Rebug Toolbox then the eid_root_key must have been extracted on a usb device, check...