Some more missed direct messages later, JammyDodger and I were able to spot a pattern 
The notification is missing when the message is a reply to another message.
An example:
These are 3 chat messages Nat sent me:
In the morning I had 2 email notifications
There is no email for the second message, which is the one that was the reply to an earlier message.
I also have a screenshot showing notifications about 2 new chat messages on the icon in the header. (I was pleased that there was no notification about badges, so I took a screenshot of my notifications. It’s a happy coincidence that the number of chat notifications is included.)
So the problem is: whenever Nat replies to another message in our chat, I am not notified 