SharePoint Online - document and folder permissions inheritance

First bullet is correct. Second bullet is not. The way the second bullet works in SharePoint is, as you said, like "how things worked in the 'old world' of file servers where moving the document from one folder to another meant that the file then had the 'protection' of the folder's permissions". That's also how it works in every version of SharePoint assuming the file does not have its own permissions.

That said, you can give a file its own permissions by breaking inheritance with the folder or library that it lives in, but this requires action. So if you don't take that action, the file inherits the permissions of its folder/library. Once you break inheritance, it has its own permissions and retains them no matter where you move the file.

Anyway, I put together a practical guide on permissions, mainly aimed at Site Owners. Check it out here: http://icansharepoint.com/practical-guide-to-sharepoint-site-permissions/

P.S. When you move a file from one folder to another you LOSE the version history of that file, so beware. Version history is one of the most powerful features of SharePoint and the only way to retain the version history if you need to move the file is to migrate the file using a third-party tool. (At least, that's the only way I know of.)