<<Perhaps best practice would be to initiate a full emergency only if a crew call for it since it could perhaps be a bit embarrassing for us if we mention to the Pax that we have a minor problem with a partial hydraulic loss and are then greeted by the blue lights when we land!!!>>
The rule - at Heathrow and Gatwick at least - is ANY hydraulic failure constitues a Full Emergency.
I'm really not at all happy with your statement above. You sound like so many pilots I've had contact with - "We have one engine u/s but do not need the safety services"., etc. Why?? Safety surely comes far above the "image" to the pax. When I fly I'd like to know that the bloke up front has my safety in mind. US pilots seem to go the other way and will, very sensibly, ask for the "services" to be out for just about any technical problem.
Just explain to the pax that there is a technical problem and for safety reasons the emergency services will standby.