Thank you for your reply @githubkoma. I have played a bit and yes, I kind of got the webhook message appearing like a private message. For this, I had to create a "pseudo user", a user which does not have any real person at the other end and add a "channel"-field in my message with the value "@pseudouser". This way, in my mattermost the user appeared under "direct messages" and the message was there.
What I miss is the "unread messages" counter (the small bubble beside the user name that indicates how many new messages there are).
The default behavior seems to be that in a channel the channel name only gets bold when there are unread messages, and only when there are mentions ("at"-all or "at"-username) then the counter bubble appears. In direct messages, this counter bubble always appears (supposedly because these are always meant like "mentioned") but in this case when a webhook bot sends messages into a "direct" channel this is still handled like a normal channel message: no way to get the counter there.
Any ideas?