I think I found the culprit. I had backed files up using cp (e.g.
10-ssl.conf to 10-ssl.default.conf) so if I made mistakes, I could revert
easily. It looks like all files in the conf.d folder are included, therefore
my backup files overwrote the standard ones.
Now, when I try to send mails, outlook tells me it can't save the message in
the "sent" folder (the mail server denies saving there (I'm translating from
French, sorry)). The error code is 0x80040119. At least, I don't get the
same set of errors.
Thanks, aki, for your previous answer.