What’s service does ATC service include?
Air traffic control service is a service provided for the purpose of:
  
  
  preventing collisions:
  
  
  between aircraft, and
  on the manoeuvring area between aircraft and obstructions; and
  
  expediting and maintaining an orderly flow of air traffic. 
  
  
  (ICAO Annex 11)


When saying a flight is provided with ATC service, it is just another way of saying that it is a controlled flight. That is, a flight where the pilot can't just do what they want, because the flight is subject to an ATC clearance; the ATCO will clear the flight to follow a certain route and fly at a certain level.

Importantly, all controlled flights are coordinated as long as they remain in controlled airspace. This ensures alerting service throughout the controlled portion of the flight. This coordination also extends to military units (so we don't have a repeat of Korean Air 007). Controlled flights are also provided with flight information service (traffic info, weather info, info on nav-aids etc.)

As stated in the quote above, ATC service is provided for the purpose of preventing collisions. However, actual separation is not necessarily guaranteed to be provided by the ATCO. The class D airspace you mention is a great example. Separation is only guaranteed to be provided by ATC for IFR-IFR conflicts. In other cases, it is ultimately the responsibility of the pilot* to prevent collisions, which they are able to do based on the flight information service provided.

*: the controller can still decide to provide separation, but is not required to do so.

Note that separation on the manoeuvring area (which includes the runway) is provided regardless of airspace class.