What people commonly refer to as the "home page" of an organization, would then be called a "welcome page," provided to welcome outsiders give a general overview. In fact, many places call this page "welcome.html." The "welcome page" of DCS/LCSR is https://www.cs.rutgers.edu/ (the default being "index.html").
Now what do you call the file, "public_html/index.html," which everyone advertises as their "home page?" In documentation I'm writing now, I try to sidestep the issue by calling it your "WWW index page" or your "public web page." If you have a customized page of starting points which comes up when you start your browser, that is really your "home page."
The whole point is probably sillier than the "cracker" vs. "hacker" debate, but I thought you might be interested. I got involved in it because index.html is not what I use for my home page. (Here is my WWW index page while here is a public version of my home page.)