A lot of companies runs private DNS servers. It isn't uncommon that companies have a local DNS for all the private IP numbers inside the company.

But that is irrelevant to the question. Any publicly registered domain includes a registration of primary and secondary DNS being responsible for the domain. And a user who needs to make a lookup for a any registered domain do not need to know the the primary or secondary DNS for that domain. The users normal DNS will - through the DNS hierarchy - get access to the information and report it back to the users machine.

Note that a private DNS sever normally gets configured with the required references to allow the private DNS to reach into the public DNS hierarchy and work as a local cache. So while the private DNS only owns private IP numbers, it will still be able to reach out and retrieve information for any publicly registered domain.

I see that you have still not looked at the Wikipedia link I gave. It mentions other companies who are using the same API.

Yes, the API is quite simple to use.