Oxeagle's initial call is the correct one per cap 413
'Birmingham Radar, G-XXXX, Requesting Flight Information Service'
You are supposed to state what you are looking for in your initial call....this change a few years ago from the old situation where you just gave your call sign.
Why do people say what they do, rather than what is correct? Oxeagle is makeing the correct call....let's not try and change that!
As for why the controller didn't ask for any additional information, could you have gotten a handover from a previous unit? If the previous controller told you to "Contact Birmingham Radar on xxx.xx" then you got a handover. That means that he has given all your details to Birmingham Radar by phone to make it easier to you.
If on the other hand you were told to "Freecall Birmingham Radar on xxx.xx" then he has not passed your details on, and you would expect to have to be asked to pass your message after your initial call.
dp