They also try to make degrees more affordable. Both Bowdoin and Trinity colleges, for example, waive application fees for first-gen students; Pitzer College has a few endowed scholarships. The University of Wisconsin just began offering free tuition for first-gen transfer students, while Duke last year created one of the most generous, comprehensive programs of all. It will select 240 first gens to attend for free all four years; they will receive a computer, books and travel between semesters at no cost.

Colleges can identify first gens on the Common Application, which asks for parents’ education history. Along with the application, the organization sends a customized summary of candidates’ attributes, including, since 2013-14, first-generation status. When this is combined with the entry that shows that the application fee was waived for financial reasons, it’s apparent who is both low income and first gen. But the Common Application definition is different from the legislative one. It’s the same as the one used by the engineering school that Ms. Weingarten called — neither parent can have a bachelor’s, even if they didn’t raise the child.

It may seem like hairsplitting. But in the frenzied competition for admission to selective schools, where counselors have seen students get their DNA tested to see if they qualify as minorities, it’s an important distinction. Several said they advise their first-gen clients to highlight their status, either in essays or interviews. When Ms. Weingarten meets new clients, one of her first questions concerns their parents’ education level. “This process is so difficult, if you have an honest advantage, you should use it,” she said.

Certainly there are affluent parents who never went to college, punching holes in the idea that first gens are by definition disadvantaged. One college counselor told of a first-gen student on the East Coast who arrived at her counseling appointment in a Porsche. She wrapped herself in the first-gen mantle, bringing it up whenever she could and was admitted to several selective schools.

Conceivably, one can be both first gen and legacy. One first-generation student in Minnesota is applying to her grandmother’s alma mater. “I definitely don’t fit the stereotype for a first-generation student,” she said. “I’m a middle-class white girl.” Her mother owns a small business and her father works as a middle manager in information technology at a Fortune 500 company.

To be sure, most first-generation students come from families with low incomes and minimal exposure to college. Only 12.5 percent of all students whose parents didn’t get a bachelor’s degree come from families with incomes exceeding $106,000, according to an analysis of federal data by Robert Kelchen, an assistant professor at Seton Hall University. Many education experts even use the terms “first gen” and “low income” interchangeably. Officials who practice “holistic” admissions — examining family background, recommendations and essays in addition to grades and test scores — say they can figure out who is truly disadvantaged by looking at how candidates overcome obstacles, whatever those may be.

How much first-gen status really matters for college admissions is unclear. At Harvard, it’s “one of 50 factors” under consideration, said William R. Fitzsimmons, dean of admissions and financial aid. “This is not a mechanical process.” Some counselors view it as a “tipping factor” for students who are tied with others in the admissions pool; others insist that it is much more important.