What is the right spelling, 乌兹别克 or 乌孜别克?
1934年，盛世才召開新疆第二次民眾大會，確定烏孜別克族為正式族名，把國外的烏孜別克族依然寫作烏茲別克。
In 1934, 盛世才 held the second public meeting in Xinjiang, determining 乌孜别克 (wū zī bié kè) as the official ethnic name, and the foreign Uzbek still writing as 乌兹别克.

Therefore, 乌孜别克 usually refers to the race in China.
But, many people still use 乌兹别克 interchangeably, as shown in the government web page.

However, 乌兹别克 mostly refers to the foreign country or race. 乌孜别克 is seldom used in this case.

I think 孜 is more popular than 兹 in Xinjiang, and that's why 盛世才 chose it. For example,
seasoning : 孜然 (cumin)
festival : 肉孜节, 诺鲁孜节
place : 克孜尔千佛洞, 克孜尔尕哈, 柏孜克里克, 克孜尔魔鬼城, 克孜勒山, 克孜利亚, 克孜库尔干, 克孜勒苏, 牙通古孜
language : 柯尔克孜语



盛世才 (Shèng Shì-cái) (1897/1/8 - 1970/7/13)
1933年至1944年间新疆军事、政治首长，有“新疆王”之称。
Sheng ruled the Xinjiang province from 1933 to 1944, and had the name of "King of Xinjiang".