Fake offline means you still connect to the server and you can even use chat. That’s basically it. This was the old offline option that was available pre-launch. With 1.0, they added a truly offline client that doesn’t need servers.
You can go to the properties of the game in LE and change this:
I’m pretty sure it doesn’t, because that was an issue before launch. But I haven’t played offline with either mode, so I can’t confirm.