Why does Entity Framework automatically use the ObjectContext instead of the DbContext when mapping database tables using ADO.NET Entity datamodel
The DbContext is a wrapper around the ObjectContext which simplifies the interface for the things we do most.
If you have an DbContext you can still access the ObjectContexttrough ((IObjectContextAdapter)dbContext).ObjectContext;
If you want to use the DbContext instead of the ObjectContext when using database first, you can switch the template that's used for generating your code. You can do this by right-clicking in your EDMX and selecting 'Add Code Generation Item'. You can then select the DbContext template.
Here is an example of the whole process.
Since VS2012 the default code generation changed from ObjectContext to DbContext.