I would call it a wall box as it is clearly mounted flush into the wall - I don't know why the LBSG call it a lamp box, maybe they have inside knowledge why Royal Mail would mount 1000s of boxes designed to be mounted onto a pole into walls.
Whether it is pillar next to a wall or a wall extension doesn't really matter.
But for the average map user, knowing they would be looking for a post box built into the wall is the important information.