Sorry for not answering your question. I am not sure, if there is a conclusive answer.
- At first, I connected the tacho-line (-> the one that delivers the RPM of the fan). The data was rubbish. This setup was only running for 30s or so - perhaps 1 minute.
- Then I googled and found out, that there has to be a connection between the two grounds. And I read about activating the internal pullup resistor (-> PWM Fan Controller - #8 by fr8tra1n). I did both at the same time. Readings were good, until a minute or so later, the d1mini died.
I don’t know. Perhaps it was pure luck, the d1mini did not die during my first attempts. Perhaps, it was the activation of the internal pullup resistor.
The last answer in that discussion points at the same direction as you do: 
Did read it, but did not understand it at the time…