The second an A/C has been identified, that automatically and immediately implies a radar service is being provided until they are told otherwise.
Not sure that I agree with that. In the terminal/approach environment there are many occasions when I have an aircraft identified but I'm not providing it with a radar service. A VFR departure may be identified on departure but unless the pilot asks for a radar service he/she continues 'doing their own thing' - but I still have them identified.
PS - I say 'doing their own thing' because in reality I'm not sure what service they are then receiving, a situation that will not be helped in the UK from March when just about anything can be called a FIS!