I've never come across any specific/official terminology to refer to the different pages. To the end user they are usually seen as the same page - the same URL - the home page.
In generic terms, the "home page when logged out" could perhaps be shortened to the "default home page" or "logged out home page" (when there is a distinct difference).
The "home page when logged in" could be referred to as the "logged in home page" or "authorised home page".
I think my favourites would be "default home page" and "authorised home page" respectively.
If this is for your own internal use, as you suggest in comments, then the important thing is to decide on a convention and stick to it. homeDefault and homeAuth perhaps?
In my opinion, a site could have multiple index and landing pages, but only one home. So only home should be used to identify this page IMO.
The homeDefault page shows the full login/register page.
The homeAuth page shows the authorised users twitter feed.
These are indeed very different pages and both accessed via the same URL (no redirection).
(DISCLAIMER: Based more on my own opinion rather than anything authoritative.)