I've spent hours researching this. It looks like what I am experiencing is the way it is supposed to work. I have to say I think it is not intuitive. I thought the aim of the new search was to prevent the users having to change contexts. But now we have a single search box that exposes multiple different user experiences. And the users will have to know to switch the context of where they are searching from in order to find what they are looking for. For example, users can't find people from the library or site. They have to go to SharePoint home or Office 365 home.
And users have to know to change the scope to get the correct filters. (File filter isn't available in the 'All' scope for example)
That seems a very surprising to me and will require quite a bit of training for the users.
Is this the intended user experience?