I was taught it with half maximum. The third image is a tricky one because the rise is not smooth, so unless someone has experience with the system (whatever it is), you could assume that the step at 1500000 counts is real or not which would shift your placement of the half maximum height.
I agree, the third image is tricky, because of the shoulder. I used this picture mainly for demonstrating the idea.
It confuses me, because in half the scientific papers they use the "edge" method, while the other half uses the "half maximum" method. For example PHI uses the "edge" method. And I saw an edge method as well in a Kratos application note.
to my perswonal experience, the method proposed by PHI works well - I was able to reproduce the work functions of different thoroughly cleaned (!) metals employing exactly this procedure ... Best regards, Dirk