Just wanted to add since I've looked into Harvard Extension School (HES): a HES graduate from a degree program is enrolled in the Harvard Alumni Association and does get to refer to themselves as a "Harvard graduate".
The degree is from the HES with a degree in "Extension Studies", which is the big distinction versus "real" Harvard, but I also suspect that beyond you first or second job, that becomes about as meaningful as your GPA.
Also importantly, that difference has come up in the last few years, to the point where the Harvard College Undergraduate Council and the Harvard Graduate Council both voted last year in favour to remove that differentiation from HES degrees[1]. It is however, Harvard University that has to decide, although it's quite interesting that the students - who are generally considered to be the ones who make the biggest fuss about "real" Harvard - are the ones in support of the move.
It's also worth noting that the child of a HES graduate is still considered a legacy, so that benefit is also conferred on their family.
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[1] https://blogs.harvard.edu/lamont/2022/03/28/harvard-college-...