It looks like it solved itself. I've no idea what happened. I would like to know, if possible, so that I can avoid it in the future.
I put the logs I found in that file.
No AV. I changed nothing since when it worked.
I played this morning. It worked well.
Then I got disconnected after putting the computer in hibernation. It didn't peep, just prevented from starting a game and displayed no public game.
I closed and restarted the client (and then the PC) to login back. The "environment" field is empty, with no choice available.
Any attempt to connect yields a "lost connection to server during login" message.
Any help would be welcome about how to fix it and how to avoid in the future.