First off, to make it clear, the question you linked at the end of your question has nothing to do with the particular feature you've referenced in the screenshot. That is about company pages, which are full pages created and edited by companies who have a Talent account with us (formerly Careers 2.0), and is used as a part of the Jobs database (each job listing is connected back to a company page for more information). Company Pages do not currently appear in that dropdown.
The feature you're referring to is currently just called "Lists" (at least that's the last thing I heard it called) and is specific to the developer story. That list of companies was originally seeded from Crunchbase back when the feature first launched in 2016 and they are not used or connected anywhere outside the developer story. There is no public mechanism for updating information for those companies, and our generic guidance for "this is wrong" has always been just remove that company from your item and create a new one with the correct information.
Over time, this has caused a lot of duplicates to be created, with a lot of companies with incredibly outdated information. There are a few solutions actively being planned out to deal with this issue, but none of them involve allowing people to edit information for the existing companies in the database. Last I heard, this particular database would be completely dropped at some point in favor of something else. But I won't go into details of those plans since I'm not entirely familiar with them.
So to answer your questions more directly:
Who curates these pages?
No one. Once they're created, they exist and are never edited.
How was this information even introduced?
Originally, data obtained from Crunchbase at the beginning of 2016. After that, whatever users typed into the boxes to create new companies.
What process does one use/do to edit fields like Company Website?
There is no process. Just click the "x" next to the company name to clear it out, and type in all the correct information over again.