Denver TRACON Controller reported a problem associated with Denver Center and how the two facilities work MEDEVAC aircraft.

Narrative

Aircraft X was a MEDEVAC aircraft arriving APA. ZDV initiated a 'Medical Emergency' for this aircraft to try to force this aircraft direct to APA. Sector 27 at ZDV coordinated a 'medical emergency' with DR4 without passing any pertinent information either to DR4 or via the phone to the supervisor. Aircraft X checked in with DR4; and the controller asked for more information regarding the emergency; and were informed by the pilot of Aircraft X that they weren't a medical emergency; however there was another actual Medical emergency on the previous frequency (Aircraft Y arriving DEN).ZDV regularly misapplies the intent of priority handling of MEDEVAC aircraft to allow the aircraft to proceed direct destination; and out of the traffic management initiatives involved at the destination airport. We aren't sure whether the Sector 27 controller confused the two medical aircraft; or they were willfully disregarding LOA procedures and TMI's. We have discussed MEDEVAC handling at length with ZDV and stated that their interpretation of what priority handling means is different in a terminal environment that it may be in a center environment. Often times; the best place for ALL aircraft; including MEDEVAC is on the appropriate STARS or LOA arrival routings. This allows D01 to more effectively and expeditiously handle all traffic in the terminal environment. This allows for a safer product for all traffic; and often provides MEDEVAC aircraft more expeditious service than direct destination.