I agree with you that it almost definitely uses more net electricity to keep the air conditioning on all day than to turn it off for part of the day, and that the website you quoted is nonsense, like you say. It's not entirely inconceivable that the website is correct--one can imagine a scenario where air conditioners are far less efficient at higher power or whatever--but I think it's unlikely (note: I don't have any specific knowledge in this area).
HOWEVER, if you're trying to be green or civic-minded rather than minimize total electricity usage, it's plausible (though still not certain) that it's a good idea to use more power overall but less power during peak usage hours. I know it differs from place to place, but personally, my electrical utility is always sending us notices begging us to reduce electricity usage at 3-6PM during hot days. In that case, keeping an AC on low all day would make life easier for your utility, reduce the need to build expensive peaker plants etc., versus letting the house get hot then turning the AC on full blast at the worst possible time. For some people, it's not just civic-mindedness: each joule of electricity may cost more during peak hours. But that depends on your particular type of electrical meter, and local policies.
But if you really want to be green and civic-minded, reduce AC usage altogether, by improving your insulation, closing shades during the day and opening them at night, opening a window or going outside or having squirt gun battles when appropriate, etc. :-D
I am asking why the cool room warms up faster (loses the cold after you turn off the AC) then the warm room loosing cool down (lose the heat after you turn off the heater).
That's not a universal thing. That depends entirely on the room and the conditions.
But in your description, one big difference is solar heating. When the room cools, (assuming you don't have significant drafts), most of the heat loss will be via conduction through the walls and convection outside. Radiative losses occur as well, but they tend to be much smaller at room temperatures.
On the heating side, the same would be true for the shaded room on a hot day. But if the room isn't shaded, you add in radiative heating. The sun is (in some situations) simply able to deliver a lot more power to your room than your room would dump on mildly chilly day.
Besides the solar heating, it depends significantly on outside temps. Not sure what part of California you are, some places have common summer temps of 100F where you have nearly 30F difference over preferred temp. You mention a heat wave where it might be another 10F above that. It might be much rarer for that location to have a day that much below temp point (say ~44F). So you might be comparing 30F heating days to 18F cooling days.
Besides the outside, you might get heat transfer from other rooms. You said this room is on the second floor. Since heat tends to rise, you might benefit from any warmth in the room below, but not much from any cooling done there.
Finally, it might be a minor point in many places, but nothing running inside your home (except the A/C) is cooling it, and everything is heating it. The refrigerator, the oven, the water heater, the dryer, the big TV, all the occupants, etc are producing heat. For a small space (especially if you're cooking), that can make a difference.
Best Answer
It depnds on the inside and outside temperature.
If the outside temperature is higher than the inside one, keeping windows open will be waste of energy, trying to cool the ouside too.
If the outside temperature is lower than the inside, a fan is enough to help cold air come in. The airconditioning itself will only work as a fan so it makes no difference in the energy spent, except the fan energy which accelerates the natural convection.