Overthinking altimeter temperature errors

Hi all, I’m trying to wrap my head around something with pressure and density altitude. I can answer all the test questions about them, and I can use an E6B to convert them. However, I get a bit confused by things like the AIM saying that there can be temperature altimeter errors and phrases like “hot to cold, look out below.” I understand that lower temperature air is more dense (which is why cold days have lower density altitudes), which therefore affects the air pressure. I suppose my question is: does this “hot to cold, look out below” mnemonic operate independently of pressure somehow? Or is it simply saying, in absence of any other information, if you go from hot air to cold air, you can also assume the pressure is higher here (because the temperature is colder)? Another question: I have gone down a few google rabbit holes where people say things like QNH is “somewhat” corrected for temperature and referencing approach altitudes in cold weather. I had previously understood that QNH is the setting which gives you field elevation when dialed into the Kollsman window, but is independent of density altitude. But [Chapter 7, Section 3](https://www.faa.gov/air_traffic/publications/atpubs/aim_html/chap7_section_3.html) of the AIM has charts that reference cold temperature error above the airfield, but not on the airfield. Is this because the airport altimeter is subject to the same cold weather errors that the plane’s altimeter is subject to?