Finally got it to work with my garage door.
I'll write how I managed, maybe it helps someone in the future.
If you're having problems with homelink, forget about the DS handbook and go straight to homelink.com. Chceck if your garage door motor and remote are on the compatibile devices list. If yes, then find the instruction (program tag) specific for your garage doors and your car (DS is not yet on the list so choose evoque).
First clear all codes in homelink by holding left and right button utill the led flashes rapidly (might take up to 20 sec). Now when you push any button it should flash slowly.
Clearing the codes is a must, although the DS handbook doesn't say so.
Next push the button you want to program and the button of your remote at the same time.
In my case the remote has a timeout function, so when you press the button it sends the signal for couple of seconds and then switches off although you keep the button pressed and the led is on. What I did was while holding the homelink button I pressed the remote button three times each time holding it for five seconds. At the third time the homelink led switched from slow flash to rapid flash and the homelink button got programmed to the garage doors.