Do pressure altimeters take into account differences in temperature?
You can't just adjust the QNH setting to compensate. The temperature adjustment is not linear. The error increases with altitude, so when the altimeter is set for QNH the error at field level is zero. But it increases the higher you go. It's not just everything being X number of feet higher or lower.

In your question you say "the aircraft's pressure altitude is lower than true altitude in winter." That's backwards. The actual altitude is actually lower than what is indicated by the pressure alt. This can run the risk of colliding with terrain if the pilot doesn't adjust. In fact, exceptionally hot weather causes an error also, but in the opposite direction. The reason this doesn't need to be compensated is because the actual altitude will be higher than indicated so it doesn't create a danger of hitting terrain.

From answers to another question an aircraft with an FMS can be set to automatically adjust. 

The problem seems to be in who's responsible for making the adjustment (ATC or pilot) and being clear on whether the altitude announced is adjusted or not. According to the answer from wbeard52, in the US the FAA has established a list of airports with cold weather procedures that require the pilot to do the compensation and announce the corrected alt to the controller. According to Sami's answer, in the EU the controller makes the corrections under radar coverage, and the pilot makes the correction without. It would appear from your question that in Korea they have decided to have you make the corrections as controller.