Do US universities publish an official definition of gender in the context of scholarships or admissions?
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.