Pressure Altitude is the height above sea level in the Standard Atmosphere (barometric pressure 29.92 in/hg, 1013 Mb). It's useful to note that absolute air pressure pressure at sea level in the Standard Atmosphere is 14.7 psia (absolute pounds per square inch).
Since the altimeter is just an absolute air pressure gauge (comparing ambient static pressure to a vacuum), this means that when the baro setting window is at 29.92, a pressure altitude indication of zero feet means the ambient static pressure at the altimeter is 14.7 psia.
Obviously, the absolute air pressure drops as you go up and increases as you go down; at 1000 ft above sea level on a standard day absolute air pressure is 14.2 psia, and if you went into a mine 1000 ft below sea level, the absolute air pressure at 1000 ft below sea level is 15.2 at a standard barometric pressure of 29.92/1013 (based on the standard atmosphere's pressure profile and lapse rate).
So think of pressure altitude as "height above/below the vertical point at which absolute pressure is 14.7 psia" wherever that happens to be, vertically speaking.
When high pressure (atmosphere is thicker where you are) or low pressure (atmosphere is thinner) weather systems are present, the level at which absolute air pressure is 14.7 psia will be above or below sea level. If a high pressure system is present, say with 30.92" of mercury, the absolute pressure at sea level will be 15.2 psia, not 14.7. If the altimeter is still at at 29.92, 15.2 psi absolute means the altimeter thinks it's at 1000 ft below sea level, and will show minus 1000 ft and you'll have to climb to 1000 ft true altitude for the altimeter to show zero.
Say the local altimeter setting is 30.92". When you turn the altimeter calibration knob to 30.92 in the window, and you are at sea level, you are adjusting the clock mechanism to make the altitude read zero feet at 15.2 psia, not 14.7 psia, to compensate for the higher pressure. Or if you are at 1000 feet elevation, where the standard absolute pressure is 14.2, you are setting the altimeter to read 1000 ft above sea level at an absolute pressure of 14.7 psia.
So you are correct about the pressure setting; there is no temperature correction for density in the altimeter setting provided. It's based on the standard atmospheric temperature of 59F/15C at sea level and the standard lapse rate going up. Due to density effects, temperature deviations from standard affects the pressure being sensed by the altimeter's aneroid chamber in a similar way to pressure deviations from standard; that is, temperature below standard has the same effect as pressure above standard, and vice versa. So when temperature is below standard, say it's near freezing, the altitude reading is higher than true altitude even when set to the correct barometric pressure, and vice versa when temperature is above standard.
However, these altitude deviations from temperature are not enough to bother with and we normally live with the error (everybody in your vicinity is following the same error, so no harm no foul from that aspect) except where the deviation becomes potentially dangerous. Potentially dangerous means the error is enough to eat into altitude margins when close to the ground or obstacles, and you can't see the ground or obstacles. When it gets really cold, the errors can get quite large, even larger the farther above above the barometric pressure source the altitude we are concerned with is.
This gets dangerous when flying IFR approaches in winter, where a 100 ft error can be lethal when precise obstacle clearance observance is critical. So when using level-off altitudes within an IFR approach procedure, and the ambient temperatures are generally at or below freezing, you have to consult a temperature correction chart and apply correction values to level off altitudes and decision altitudes within the approach.
Outside of that circumstance (and sometimes enroute altitudes, if cruising close to minimum enroute altitudes in extreme cold where obstacle clearance is a concern), we don't normally bother with non-standard temperature corrections to altimeter readings, only the effect of high temperature related density reductions on takeoff performance.