Scott,
With your Office 365 subscription you also get the Office 2013 Suite which allows you to build desktop applications. If you are using Access 2013 then you are using the desktop application. Only if you selected a web database would you be building one
in Access 365. Now, the fact that is was built for you and you ported it over to your machine also means this is a Desktop database.
And yes, Access 365 is web based only. The other applications allow you to create local copies. If there is a way to create a local database with Access 365 I have not found... yet! Summing up, you have a Desktop database and are using Access 2013.
Any clearer now?
Thanks Gina, you have explained it very well.
I will keep trying to overcome my problem and let you know if I succeed.