Sadly calling them only resulted in the answer that the free visa is availably during booking. I already have tickets for the whole family, and it seems I would need to buy evisa or VOA if I wanted to leave the airport (costly sadly).

I also asked if I could make a booking om the same date, include free visa, and then cancel the booking, and although I couldn't see the person I was talking to, I am sure she was frowning upon me. Afterwards I received a link to the visa section of saudia, which did not include anything free.