Uhm, doesn't that entirely depends on the washing machine? I don't see how they come with such results. Are they average numbers or something? If I see how much water I use they should be really really water efficient.
It assumes a modern dishwasher. Energy standards vary, but not wildly for dishwashers I think. If your dishwasher is super old your mileage may vary.
I have experience with 3 dish washers, not too old and I am sure they use more water then I do washing by hand. I have not measured it though but it has got to be more than what fits in my sink to clean a full dish washer. The difference in the amount of soap/detergent is also doubtful. You don't need much for a dishwasher but for handwashing counts the same. Especially if you let the dishes lay in there for awhile roughly 80% is already clean.
I either use the dishwasher now, or the husband. But when my daughter was born I went a little (ok a lot) nutty about germs, dish washing was like a military operation and no one else was allowed to interfere. My hands were not a pretty sight back then.
I wash up as I cook. When I serve the food from pots and pans onto serving dishes I place them in a sink of hot soapy water. Once the meal is completed, I change the water and wash the pots and pans. I run clean water and place all the dishes in to soak for a few minutes. During that time I clean the stove top, table, counters. I then prewash the dishes and place them in the dishwasher. I usually run the machine nightly as it tends to be full after a full day.