Most of us ( me anyway) would expect a construction crane to have a vertical support and a horizontal gib just above the building under construction. This one appears to have had a gib which extended upwards at an angle. The pilot may have seen the bulk of the building and the vertical support and assumed it safe to fly above them, not seeing the gib in the poor visibility. It probably wouldn't matter if the gib was lit as it wouldn't be seen in the cloud anyway. He probably thought that he was safe as he could still see the ground albeit from the cloud base.
You raise a valid point. The NOTAM doesn't state the type of crane. Merely that there is one there. Although the NOTAM by all accounts does state the maximum height of any obstruction in that area. So theoretically it was still noted that there was an object of maximum height xxxft in a xx nm radius of xxxx
Which ideally would have been the height of the jib raised to its maximum vertical position