If the President (doesn't matter who) showed up at your house, in your living room, would you still claim the right to deny entry to anyone you chose?

The judgement governs the President's use of Twitter, not the interaction between Twitter and other people. Twitter does not magically become a Public Forum, just because the President uses it.

Now, I'm not sure that is Right and Just, but I'm also not going to worry about it too much. I don't use it enough to get completely wound up about it.