I would take that as it literally reads, which is that you have to be departing on, or arriving from, your flight in the Delta One cabin. In this case, the passenger was not. He was, instead, departing on a non-Delta One flight and connecting later to a Delta One flight. I think this would typically allow entrance to a SkyClub at the originating station, but the way the website is worded, it would not allow Delta One Lounge access. At least, as I read it.