JD I agree entirely with your para 2, but I got rapped over the knuckles by the SATCO a couple of years ago for doing that. He said that as I was providing an Approach Control Service I should call myself 'Approach'.
The main problem in the Approach/Radar controversy is that there is no guidance as to whether the unit's callsign should reflect it's function (Approach), or the equipment used to provide the function (Radar). So a controller identifying himself as 'XXX Approach' may be using radar. But then again he might not. Similarly, if a controller identifies himself as 'XXX Radar' he may be providing an approach control service. But he might not. As usual, the traditional remedy of looking it up in the MATS 1 only muddies the water. What exactly are 'ATS Surveillance Services (in general)? And isn't Area Control an ATS Surveillance Service? If so, why is its callsign 'Control' and not 'Radar'?