This is Andrew Stegmaier from the Access product team. I wanted to clarify some of the issues that you have raised.

Access 2013 allows you to make two kinds of databases (1) Web Databases and (2) Desktop Databases.

2013 Web Databases are published to a SharePoint 2013 site as a SharePoint App, and the data is stored on the back-end in SQL Server. These were the kinds of databases that you've been reading about in these linked articles. We're really excited about the potential of this new kind of database to make sharing, deployment, and creation easier than ever. Office 365 means that its easier than ever for people enjoy the benefits of SharePoint without expensive servers and IT staff.

However, as Felix9 pointed out, Access 2013 supports Desktop Databases as well. These are the kinds of databases that could also be created in previous versions of Access, which by default store the data in the Jet database engine. (Just as before, these desktop databases also have the ability to connect to SQL for data storage, as well). Access is still a very useful product by itself with no Office 365 or SharePoint.