My, what rotten luck, and I'm sorry you had such a bad result.
Looking at your photos, I believe the real problem is the abrupt transition in the adapter socket where it meets the square drive portion and
possibly the materials/means of manufacture. Though I don't know what torque it was pulling when it failed, it can't have been much overall, and it appears the drive adapter was of poor quality despite the brand -- the metal appears crystallized. Usually, when a forging of this sort fails, there is a true twist to the shank. I have at times needed to overuse such adapters to remove frozen fasteners and have twisted them a good 18-20 degrees off- center and still did not fail as yours and one of mine did. I have gotten the occasional 3/8"->1/4" drive adapter that had been cast and not forged as claimed
. It happens surprisingly often in that size, because the manufacturers (wrongly) figure it will never be subjected to much torque. Though it occurs less often in 1/2"->3/8" adapters, I have seen it (and the results) firsthand. I had a similar experience once -- and once was all it took to go to a different brand of adapter. End of problem.
The hex-key socket can be salvaged by drilling and tapping the remaining slug of the old drive adapter, mounting the hex-key socket tightly in a padded vise, and using a slide hammer to extract it.
The ultimate solution to prevent future problems is a new drive adapter. Here in the States, I would suggest Snap-On or S-K. Where you're located, Facom have an excellent reputation: http://www.facom.com/fr-en/h-30-Facom.html
If you could salvage your damaged hex-key sockets and simply replace the adapter, you'd be all set with the rest of the tools you already have. If you can't find something where you are, let me know and I will be happy to look for something here. 8mm and 6mm hex-key sockets can be obtained separately as well.
Excellent photos, by the way.