Because the post advises you not to give your key to a friend, I think you are supposed to get an actual key to redeem through Steam. Click around in your account history and see if you can find anything like that. I'd attribute any errors atm to the sheer server load.
Personally I would wait until much later today to try and redeem mine just to avoid tempting fate.
The top post said:
Click ‘Claim an Elite: Dangerous’ key.
You can redeem your code by selecting 'Games > Activate a Product On Steam' in the Steam UI.
I've never been offered a "Claim an Elite: Dangerous key" thing to click, only "link to Steam". We're missing this part.