High School Research–does where you research matter for admissions?

Hey guys. I’m trying to get a research internship at a university, so I was wondering if the prestige of said university would impact admissions. For example: Would research at like Stanford be better for college applications than research at a less “prestigious” school?

Thanks. I’m sorry if that comes across as pretentious.

From what I understand, how deep you get into your research, how far you take it, how much you contributed, and how it ties in with your passion is more important than the institution.