Hi,
well, I had already tried to see if a new account would show the same issues, and it did, so this didn't seem to be the right path to follow. Before that, I even tried resetting the keychain, as suggested in another post about similar issues, but that didn't work either. Apple support just asked me to delete and add some accounts under both Message and FaceTime, but that led to nothing, too.
As I've said yesterday, the fact that, after three clean installs without any kind of backup restoration, neither any of my previous AppleIDs nor any newer AppleID created from scratch was able to activate both apps, to me meant that there was something wrong outside of my Mac. Today, after my latest phone call to support, this has been confirmed to be true.
Yesterday I've been lucky enough to chat with a senior advisor who then left some useful notes for today's phone call, and today, after insisting again for a few times to talk to another senior advisor, I've found another one who knew exactly what to do. Two were the possible explanations for such kind of error messages:
[1] - Something wrong might have happened to the security info of those AppleIDs, and they would need to be modified locally or remotely, or unlocked, or reset;
[2] - Something wrong might have happened to the security info of my Mac, and this would need to be modified, locally or remotely, or unlocked, or reset.
Given the fact that all my AppleIDs were able to activate both apps on other Macs at the Apple Store, the first scenario was out of question, but the second was much more interesting.
Two years ago, indeed, one of our local Apple Store had to replace the motherboard twice in just a week, because the first replacement had already some issues with the famous Nvidia chip. They registered the serial correctly on the new motherboard, but apparently something wrong happened on the back-end servers at Apple's, and the security info about that serial number was not (entirely) correct. Strangely enough, that didn't cause any issue under Snow Leopard or Mountain Lion, but the issue was triggered by the install of Mavericks, who wasn't able to deal with what it found on those servers.
I have no details of what exactly was wrong, but the senior advisor knew exactly what to do, and even if he knew just the 'Customer Code' of one of my AppleIDs, he managed to 'unlock' all the others, by correcting the anomaly on their databases.
So, at least on this case, it has been useful to:
1] Have a clean install of Mavericks - it gave me and Apple a way to be sure that there wasn't a particular third-party app causing the issue; not so important, maybe, but it also granted me support without paying any fee;
2] Check whether it was an issue related to a bad configuration of those AppleIDs, on the appleid.apple.com website;
3] Check if those AppleIDs had or not the same issues on other Macs;
4] When talking with Apple Support, insist, insist, insist that the 'Customer Code' of those error messages wasn't something to be overlooked, and try everything possible to get in touch with a senior advisor.
Then, as you said,
"With this particular message it can only be solved by using the Contact number."
One has just to be very patient...
Thanks a lot to everyone who has experienced this issue and has posted here details, configurations, workarounds and advices, and a special thanks to the two Senior Advisors who have so brilliantly helped me after two weeks of troubles !!
Monica