Shouldn't turning the air conditioner off during the day save energy?

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