I am a little confused on the code first entity framework database.
I created a new DbContext and class that I will be storing in that context, like this:
Yep - based on the parameter in your defaultConnectionFactory it looks like you're using the default SQL 2014 instance, not the 2012 instance.
The (localdb)\ProjectsV12 instance is created by SQL Server Data Tools (SSDT)
(localdb)\MSSQLLocalDB is the SQL Server 2014 LocalDB default instance name
And (localdb)\v11.0 is the SQL Server 2012 LocalDB default instance name