The government’s filings came as part of a fight over a question that has arisen over and over in the case: What is the distinction between a legitimate donation to a university athletic program and a bribe?

Ms. Loughlin and Mr. Giannulli, who prosecutors say paid $500,000 as part of the scheme, are scheduled to go to trial in October, and a federal judge in the case offered no indication that he would delay the trial because of the coronavirus pandemic. It is to be the first trial in a sprawling case that involves accusations of cheating on college admissions exams and bribing college coaches to recruit students as athletes based on false credentials. More than 20 parents have already pleaded guilty in the case and 16 have received sentences, ranging from no prison time to nine months.

Lawyers for Ms. Loughlin, Mr. Giannulli and other parents, who have pleaded not guilty to the charges, have repeatedly argued that their clients believed they were pursuing a legitimate route to gain their children an advantage in the admissions process at U.S.C. and other colleges. They were donating money to specific programs in the athletic department at Mr. Singer’s behest, the lawyers have indicated. And they have pointed to evidence that the U.S.C. athletic department had a pattern of advocating for applicants whose families had made donations.

But prosecutors have said that the parents knew that their children were being falsely presented as athletes in exchange for their payments, and that, whether the term “bribe” was used or not, the parents knew they were engaging in illicit acts.

In recent weeks, some parents have asked for dismissals of their case, accusing the prosecutors of twisting the facts in an attempt to turn legal, if unsavory, practices into crimes. Their argument hinged on a note written by Mr. Singer, who has pleaded guilty to orchestrating the scheme. In the note, written in 2018 on his iPhone shortly after Mr. Singer was confronted by government agents and agreed to cooperate, Mr. Singer said the agents were trying to get him to “bend the truth” in recorded phone calls with his clients.