Is vertical separation based on geometric difference or altimeter difference?
The short answer: "Altimeter difference"



The long answer:

From a pilot perspective, when issued an altitude to maintain by ATC, he/she will fly the altitude in reference to his/her altimeter, whether or not it's a "Flight Level" assignment (usually FL180 and above in North America), or "altitude" assignment (below FL 180). 

For example, when the temperature is colder than ISA and one aircraft is assigned FL250 and another is assigned FL260 , the  actual (geometric/true) vertical separation between the aircraft would be marginally/somewhere less than 1000 ft. There is no ATC correction (in the U.S.) for actual "true" altitude being (probably slightly) less than 1000 ft. in this scenario.

Since there is not an automatic correction for colder than ISA (or warmer than ISA) internal to the altimeter, true altitude is different than what is being displayed to the pilot on his/her altimeter and being displayed to the controller on ATC radar. 

This is only with respect to separation between aircraft.  Separation from terrain is considered both by the pilot and ATC, as necessary.  For example, at some locations during very cold temperatures, MVA (minimum vectoring altitude) for a specific area may be raised so that aircraft always have the necessary "true" altitude separation from the terrain while being vectored by the controller.

The pilot, while conducting instrument approach procedures at specifically designated airports, will make altitude adjustments for the IAF/IF/FAF/MDA/DA/missed approach holding, based on a correction value obtained either automatically from the onboard equipment, or manually from a cold temperature altitude correction table/chart.

On these occasions, the pilot has the obligation to notify ATC of the altitude adjustments being made. These adjustments will be made to the indicated altitude being flown for the purpose of obstacle/terrain clearance and the indicated altitude on the altimeter will be higher than published on the chart.