You do not need to say the ground station name on any other call than the initial contact.
As far as "position" goes, it is normally used in non-radar enroute environment with poor or non-existing ATC-to-ATC communication, where you need to pass a lot of information. It goes something like this:
ABC123: "Control, good morning, ABC123, position"
Control: "ABC123, good morning, go ahead"
ABC123: "ABC123, overhead ABC, FL370, estimating DEF at 1205Z, GHI next, estimating destination ZZZZ at 1310Z, registration G-ABCD, endurance 4 hours"
The amount of information you are required to pass depends on the FIR, the above is just a random example.