Dear Plot-Paris,
Yes, that all makes sense. What is confusing is that after using the off-set tool one can click on individual areas and see that they all separated by circle entities. So intuitively, one assumes that on the next push-pull the circle entity will divide the cylinder, which it doesn't. Pulling the the outer annular area first solves this problem, or doing what Alan suggests, intersecting before deleting
Another mystery solved.
Many thanks,
Bob