Thank you @Mahko_Mahko and @ssieb for your work.
I can confirm that this works on the (new) version of the BOHO OFFICE Basic Line desk.
I can also confirm that I could only get it working by using Pin D1 (5) on my ESP8266 D1 Mini.
I could not get it working by using Pin RXD (3).
I also got @dev0 method working (minus button 4 because it does not exist on my desk) and I plan on controlling the buttons via their method and get the current hight via @ssieb custom component.
The only thing I am still trying to figure out with @dev0 method is how I can e.g.: press the button “Up” with my mouse for a long(er) time which results in the button actually being pressed for a longer time. So that the behavior is the same as when I keep pressing down on the actual physical control panel button with my finger.
(image of the control box of my BOHO OFFICE Basic Line desk)