I was not aware of the requirement to make a second passing level call. I always thought that the initial SSR check was enough. I don't remember ever being asked for a second passing level check, so one could presume that it is not necessary.
After giving a level check call to STN DIR, giving another to London on the next call is not required. If it was, we would be asked for it by London. I think Kilte has got this wrong. I can't find the requirement for a second call in CAP 413.
'The First Enroute ATS Unit' in this case must be STN DIR. It depends what you call 'enroute'. For the a/c, enroute is anwhere betwenn take off and landing.