When I checked the compatibility list on homlink.com I realised that my particular model is not in the list of compatible garage door openers... bummer.
I went and looked for solutions online and ended up ordering the universal receiver from homelink directly. Once it arrived the instructions said that one should press the three buttons on the homelink remote in the car - which the Tesla obviously does not have. After calling their support they told me that I also need an original remote control, which can also be ordered through their shop. Yesterday this remote arrived as well and I was able to install everything in my garage.
Universal receiver installation
Learning the codes and configuring the model S
After that I just went through the Model S homelink setup again and it immediately worked. Now when I drive up to my garage, the homelink button pops up and I can open the garage with one tap of my finger. I originally thought that the garage would open automatically once I approach it, but so far this does not happen. I guess it might be a safety feature. Imagine someone is parked in front of your garage when you drive up : )