The first volume is the correct one. I guess the question really is why would you expect the volume to be the "mean" area of cross sections time the height to give the correct volume.
(I looked up "mean area method" "volume" on Google and got references to two books, one on surveying and the other on "Railroad field geometry". In both the "mean area method" was an approximation to true volume got by using the area of a number of cross sections at different points. The second also referred to the "average end area method" which just used the areas at the two ends and is, at best, a "quick and dirty" estimate.)