Who are alumni?

Are alumni graduates or students? It appears Universities disagree on this issue from what I’ve read.

Alumnus

noun (Plural alumni)

A graduate or former student, especially a male one, of a particular school, college, or university.
‘a Harvard alumnus’

I always thought an alumnus or alumna was a graduate and not just someone who attended for a period of time and either transferred or dropped out and certainly not a current student. The above definition is incorrect, in my mind.

Including former students (who didn’t graduate) is considered by many colleges to open the pool of contacts, for all sorts of reasons. But for pull or legacy, some college will expect the parent (or whatever) to have completed their program.

That’s what I thought too, but Harvard still includes Bill Gates and Mark Zuckerberg on their alumni list.