I think half the reason no one talks about this consistently is there's no naming consistency from the media.
Is it Uighur or Uyghur?
Is it Uighur or Uyghur?
Anyway I think Uyghur is more widely used one
> “Uighur,” with an “i,” has appeared for centuries in writings by Western scholars, and many Western media and experts on the region still prefer this spelling.
However, apparently native speakers prefer the other spelling:
> But members of this mostly Muslim ethnic group overwhelmingly prefer the spelling “Uyghur,” which they say more closely approximates the proper orthography and pronunciation in their native language, “Uyƣur."
The use of the latter spelling in the West seems to be a more recent shift to accommodate this.
Source: https://www.rfa.org/english/news/uyghur/uyghur-spelling-0906...