I was aware of Hungarian names typically using the family name first, but as far as I can tell, this rule is not adhered to it in sports; that is, I have never seen a Hungarian sportsperson whose name has been used in the proper Hungarian order. It's the same with Japanese names -- Japan uses the family name first, but the order of first/last name seems to be more commonplace in sports. I have, however, seen the proper name order used for Chinese, Taiwanese and Korean players.
I agree with you, more Chinese people introduce ourselves to Westerners in the Eastern order, but maybe Japanese don't do the same. I can't impose the way Westerners call us, coz they might not know. But some Chinese TFers here still calling Na Li or Jie Zheng really piss me off. A quarter of world's population use this custom, but I feel it's not respected enough, even by some of ourselves. In contrast, Western names have never been translated into Chinese in a reversed order (family name ahead of given name) since 1930s.
As to Hungarians, the situation is different. Magyars have moved from the East to Europe for more than 1000 years, and have been mostly Westernized, I guess they already call themselves in the Western way quite often.
Btw, about Zheng Saisai. She shares the same surname with Zheng Jie, but that actually comes from Saisai's mother not father. Her father is ethnic Tibetan and Tibetan people don't have family names, so she took her mother's. And she also has a Tibetan name "索典卓玛", which I don't know how to write in latin letters.