Do US universities publish an official definition of what constitutes a particular race/ethnicity in the context of scholarships or admissions?
The short answer is no. There is no definition. Our history is very complicated around race and after several hundred years we are still grappling with it. There may be a bit of abuse of the system, but not enough to cause people to try to start defining one another with rigid rules that we would not accept in any case. We have enough problems without that. 

In places where such definitions have been used, some of them lead to atrocities. There are current examples in the world now, actually.