O2 Integrator value is a real-time value that's continually updating based on lambda sensor output IF the engine is in closed loop running. It allows a relatively small range of adjustment to occur instantly based on engine operating conditions.
The other values are long term values that are calculated based on the average Integrator value when running in that phase of engine operation, and stored in non volatile memory. In other words, if your integrator is averaging 135 at idle, the idle trim is increased so that every time the engine idles, that value can immediately be applied to get the engine running close to correct, and the integrator will then offset from there to make any small adjustments required.
Long term values are more about correcting for long term changes in the system - wear, small changes between fuelling accuracy at different altitudes, summer and winter weather, etc.
Of course if an LPG setup is slightly out in a particular combination of engine speed and load, this will, over time, cause the long term trim for that phase to wander off.