I believe that Approach is the correct phraseology. First, have a look at these definitions from the Pilot/Controller Glossary (my emphasis):

APPROACH CONTROL SERVICE- Air traffic control service provided by an approach control facility for arriving and departing VFR/IFR aircraft and, on occasion, en route aircraft.

DEPARTURE CONTROL- A function of an approach control facility providing air traffic control service for departing IFR and, under certain conditions, VFR aircraft.

The AIM 5-2-7 also says that Departure Control is for departing aircraft only:

Departure Control is an approach control function responsible for ensuring separation between departures.

So according to the FAA's definitions, only Approach Control has any responsibility for en route traffic, which includes the scenario you described in your question.

Second, note that the VFR sectional chart always requests en route pilots to contact Approach, not Departure:

FWIW, I was also taught that "Departure" is used only when actually departing a field and that all 'pop-up' communication with the TRACON should use "Approach"; this is in line with the information above.

I believe that Approach is the correct phraseology. First, have a look at these definitions from the Pilot/Controller Glossary (my emphasis):

APPROACH CONTROL SERVICE- Air traffic control service provided by an approach control facility for arriving and departing VFR/IFR aircraft and, on occasion, en route aircraft.

DEPARTURE CONTROL- A function of an approach control facility providing air traffic control service for departing IFR and, under certain conditions, VFR aircraft.

The AIM 5-2-7 also says that Departure Control is for departing aircraft only:

Departure Control is an approach control function responsible for ensuring separation between departures.

So according to the FAA's definitions, only Approach Control has any responsibility for en route traffic, which includes the scenario you described in your question.

Second, note that the VFR sectional chart always requests en route pilots to contact Approach, not Departure:

FWIW, I was also taught that "Departure" is used only when actually departing a field and that all 'pop-up' communication with the TRACON should use "Approach"; this is in line with the information above.