I was a student in 2020, and instead of getting money directly from the IRS during the first stimulus package, I got money from my school. Does it count as stimulus money

Not sure what you received from your school but it could not have been a stimulus check.   Stimulus checks were issued  by the IRS--not schools.    If you were a student in 2020 and have been claimed as a dependent for 2020 on your parents' tax return you are not eligible to receive a stimulus check.  Dependents are not eligible.  Did you say on your return that you cannot be claimed as someone else's dependent?

Q. When filing my taxes, it asked me if I received $1200 for the first round of stimulus. I didn't receive any money directly from the IRS until the second payment. Should I say yes or no?

A. Answer no.  The Emergency Financial Aid for College Students* is separate from the stimulus payments that most taxpayers got. 

Under the CARES Act, if you are claimed, or qualify to be claimed, as a dependent on someone else’s 2019 return you cannot receive a stimulus check, in 2020.   If you qualified as a dependent for 2019, but will not be for 2020, you will most likely get it in 2021, when you file a 2020 tax return.

Note that the requirement is not just whether you are actually claimed as a dependent, it's whether you qualify to be claimed as a dependent. 

*https://www.nerdwallet.com/article/loans/student-loans/emergency-food-housing-college