What is the difference between Organization and NewsMediaOrganization?
The type NewsMediaOrganization represents the organization of the news publisher and is a more specific type from the type Organization. Check the description of Schema about this type:
A News/Media organization such as a newspaper or TV station.
We publish news on our site that are yet rarely but still get featured
on top stories. We use Organization in our schema markup. Should we
change it to NewsMediaOrganization?
This is not a requirement, but it can help to represent your organization as a news publisher.
Moreover, is it still true, that Google says, that Organization markup
should remain only on home page?
This is completely false information. Google has the following requirement for structured data that you want to show in rich results:
Your structured data should be a true representation of the
Thus, the answer to your question "Same story - Article or NewsArticle? Or even both?" maybe the following:
If the main content of the web page is a simple article, then use the top-level type such as Article.
If the main content of the web page is a news article, then use the top-level type such as NewsArticle. However, using a top-level type of Article for a news article will not be a mistake, but it will have more general significance than a more specified type such as NewsArticle.