Whatever university has the strongest claim to be the oldest currently existing university probably also counts as the first university, because of the irregularity of the structure of learning institutions in the Classical and pre-Classical era (to the best of our knowledge).

That's a weird argument. So if the University of al-Qarawiyyin stops operating, then some other place becomes the first university?

Generally we consider a university more like a school - with set staff, set students, and a set course of learning

That's more of a modern understanding of what a university is like (and I doubt it is an accurate description of the mosque that Fatima al-Fihri made).

Academy was a place where philosophers and important folk got together to mutually intellectually enrich each other,

A 'community of masters and scholars' one might say, or in latin 'universitas magistrorum et scholarium' which is the origin of the word 'university'. The original point of Universities was the same as that of the Academy. The teaching of students is just a side effect of achieving that goal since teaching is a great way to learn.