The “no standing on the toilet” sign is because in Asia, most of their toilets aren’t the traditional western stool toilets, but rather, just a porcelain hole in the ground. (e.g. http://scott.sherrillmix.com/res/images/China_toilet.jpg)

So you’ll see a lot of country folk standing on the rim of toilets in the bigger cities.

True. When I was working at my last job, we had a Chinese lady from the villages who clearly used to climb up on the porcelain, after lifting the toilet seat up, to do her toilet. You could see the footprints she left on the edge of the pan. None of us had the heart to tell her, and she wasn't doing any harm so we let her get on with it.