Do US universities publish an official definition of gender in the context of scholarships or admissions? Let's say that a person assigned male at birth identifies as a woman. At which stage of their transition process would they be allowed to claim themselves as a woman in the context of college applications or scholarships? Is merely identifying as a woman sufficient? Or maybe legally changing your gender is required?

Or if we look at it the other way around: At which point would a person assigned female at birth be required to identify themselves as a man in college applications?

Note that this question is of practical value as there are presumably hundreds of thousands of people in the US who identify with a different gender.
Virtually all universities will follow the policies regarding gender identity that are promulgated by the U.S. Department of Education.  To do otherwise would mean putting their access to federal funds at risk, since Title IX of the Civil Rights Act bans discrimination on the basis of sex by any institution that receives federal government assistance.  Many institutions have an official statement somewhere (on their Web site, in the course bulletin, etc.) stating that the abide by these federal guidelines.

The Department of Education enunciates their interpretations of the law through what are known as "Dear Colleague Letters."  The ones covering the question of how gender identity is defined may be found here, on the agency's Web page.  In light of the current political situation, the Department of Education's rules cannot be seen as stable, at present, but the current state of guidance can be found there.