From the article, you can see that for a very long time Muslim were known as the Mohamed followers and only in modern times the Muslim names takes over.
The scripture is an open document, is the foundation. Different religions grew out of it by taking a position on what's the key thing. Christianity works by adding new testament and make Christ as the key thing? Mormon added a new scripture book of Mormon to make Mormon the key thing? Muslim by adding the Koran and make Mohamed the key thing? Judaism sticks with the scripture but it takes the Jewish race as the key thing?