I realise that in your part of the world, "school" can also include higher education. However, it doesn't in other parts. Here in the UK, for example, "school" means "primary and secondary education".
- An institution for educating children: Ryder’s children did not go to school at all
There are a number of other definitions on that page. I believe the one that is confusing you is this one:
2.1 North American • informal another term for university. Harvard is certainly not a loafer’s school.
Note that this specifically says that it is both North American and informal. In the rest of the world, talking about "going to school" or "school students" means the education system for children.
It's also worth noting that here in the UK at least, if you just said "students" you would be referring to university students. We don't tend to refer to schoolchildren as students at all - if we did, we'd specifically say school students to avoid confusion. Saying schoolchildren or pupils is much more common.
The following conversation I overheard a few years ago illustrates the confusion that using "student" to mean "school student" can bring.
(Cultural context: A-levels are qualifications gained at the end of secondary education, at age 18. You take 3-4 of them. Whilst you can do a degree in more than one subject, if you did it would be "Maths with Chemistry" or similar, not just a list; we don't "major" and "minor" in subjects.)
"What do you do?"
"I'm a student."
"Oh, right. What are you studying?"
"Maths, Chemistry, English and French."
"Oh, you mean you're an A-level student!"
(I'd wager the reason the second speaker even said "I'm a student" instead of "I'm still at school" to begin with was probably because "What do you do?" suggests the first speaker was expecting a job as a reply and second speaker was probably a little self-conscious about saying, "I'm still at school.")