I don't really understand what remains unclear. The cited section explains that when Java got put into the browser, executives of both companies got together and decided that it would be good PR to rename LiveScript into JavaScript.
As Java was a trademark by Sun, they obviously had to register JavaSript as well (if they hadn't already done so - I did not look up the timeline, but it's not really relevant to the argument). Getting a license from Sun made it possible for Netscape to use the name without the Java trademark becoming diluted. Remember, if you don't defend your trademark, you'll lose it.
True, but for a long time the only noteworthy "other user" was Microsoft and they kept calling it JScript for years even after they adopted the standard.
I'm not sure what exactly happened, but at some time after or during the lifetime of IE9 they stopped referring to their dialect of ECMAScript as "JScript" and began calling it "JavaScript". The most recent stuff I can find about JScript is about JScript.NET, which seems to be discontinued.
As Google, Apple and Microsoft are openly using the JavaScript trademark (without marking it as a trademark) these days, I'm going to go out on a limb here and guess that it's okay to use the term "JavaScript" to refer to an ECMAScript implementation that is compatible with the original JavaScript language developed by Netscape. This would probably preclude Adobe's ActionScript and other "enhanced" flavours of ES (e.g. JSX? I've never seen Facebook refer to JSX as "JavaScript <something>" in a way that would imply JSX is simply JavaScript).
As Java was a trademark by Sun, they obviously had to register JavaSript as well (if they hadn't already done so - I did not look up the timeline, but it's not really relevant to the argument). Getting a license from Sun made it possible for Netscape to use the name without the Java trademark becoming diluted. Remember, if you don't defend your trademark, you'll lose it.