Guys, if you think there is a mistake on the site and express your opinion, then provide sources where this is indicated.

Thank you for this website, but it seems that you have taught your students wrong thing in this lesson. Within the "ura.nis.edu.kz", the edu.kz - top level domain, nis - second level domain (and also the domain name), www.ura - subdomain (it is because www.ura is before the domain name)

I agree only with the subdomain "ura"

Ok, thanks!
If you are ok withthat, I wanted to ask you: is it correct that lower-level domain is created
within the upper-level domain? For example, within the host “nis.edu.kz”, the “ura”
domain is created for the NIS Uralsk. Or, within the host “edu.kz”, the “nis”
domain is created for NIS schools. Similarly, if a new NIS Temirtau is opened,
its school officials will have to ask the “nis.edu.kz” host to create new
domain “tem.nis.edu.kz” for them. Is it right?

Yes. Moreover, if NIS Uralsk creates its own project on its own host, its address will be new_host.ura.nis.edu.kz/

If my understanding is right,then I agree now that when both country’s domain and generic domains are used,
the former becomes a top-level domain, and the latter is second-level. It is
clear because often, the country regulates the educational organizations within
itself, so to create a new school domain, like “nis.edu.kz”, NIS system has to
ask local host “edu.kz” to create new domain, not international host “edu”. It
means that the top-level domain, i.e. the “main one”, which everybody has to
ask to create a new domain, is only “kz”, and “edu” becomes second-level domain.
This seems to be true for most of the countries apart from USA.
 
 
To understand the levels
better, the figure 7.1 on page 650 of Tanenbaum, Computer Networks (https://kr-labs.com.ua/books/Tenenbaum_KS.pdf), helped me a lot. The top-level domain consists ofonly the uppermost hosts (edu, xxx, kz, us). For example, in the 8th example in
that figure, “uwa.edu.au”, the uppermost host "au" is the TLD, and
then the level of domain increases by one, as they go down. Very
interesting, thanks.

У Таненбаума серьёзная книга. Одна из первых полных книг по сетям

But regarding the subdomains,I am a bit puzzled. In the Wikipedia page about subdomains, it is stated that
any domain within the domain name can be both a host and a subdomain. I
understand this structure, as in “www.ura.nis.edu.kz” the domain “nis” is
simultaneously hosting the “ura” domain, and is a subdomain for “edu” host.
Similarly, a father is simultaneously a father for his son and son for his own
father.
 
So, terms “host” and “subdomain”,
as I understand now, denote the relation of the domain towards other domains near
it. The top level domain is the one with no host: it is itself only the host
for other domains. The second level domain is the subdomain for the TLD, but host
for the third level domain, and so on.
 
But then, it turns out, the
term ‘subdomain” has a second meaning, which is “the lowest domain”. And it is
very ambiguous. Let’s say I opened a website washington.edu.It would be natural to assume that “washington” is the subdomain, but in
reality, it hosts many other domains such as “cs” from “cs.washington.edu”. So
now, when I am on the first website washington.edu, I know that it has many
other smaller domains, but currently the smallest domain is “Washington”
itself. Which is subdomain now?
 
 
Also, regarding the “www”:
In this example
(https://superuser.com/questions/60006/what-is-the-purpose-of-the-www-subdomain)
is is seen that "ftp.example.com", which existed before the
"www", denoted the ftp server of the host "example.com".
The author of the comment says that "www" was a similar subdomain,
which denoted the WWW server. So it is even lower in rank than the next
standing domain. Thus, it is also natural and intuitive, as well as correct, to
assume that “www” is a subdomain.

Поэтому префикс www постепенно уходит в прошлое и теряет актуальность.

Благодарю за ваши комментарии и потраченное на них время!

я с вами полностью солидарен

isnt www.ura is sub domain not 4 level because it comes before domain name nis?

edu.kz isnt top level??????????????

​The domain "edu.kz" is not a top-level domain (TLD); instead, it is a second-level domain under Kazakhstan's country code top-level domain (ccTLD), ".kz". In Kazakhstan, ".edu.kz" is designated specifically for licensed educational organizations. This structure is common in many countries where educational institutions use a second-level domain like ".edu" beneath their ccTLD.

so, do I understand it correctly:
In harvard.edu, the "edu" would be gTLD, but in www.ura.nis.edu.kz it would be second level domain?

Correct