“According to this data, the plane’s altitude was only fluctuating by about 20-30 feet. This is more stable than I expected!” Autopilots are very good and they are servoing to the pressure altitude. Many pressure altitude encoders used in modern aircraft (for example to drive altitudes that transponders report to SSR radar or via ADS-B) have 25 ft encoding resolution. That 25ft resolution is likely what is being seen…
It would have been cool to use a phone to record a GPS track with altitude and compare them. Pressure != GPS. Also wonder if there would be distinct jumps in the difference if they reset the pressure based altimeter to a different AWOS.
Not sure how it works in big planes, but in little ones you need to set your altimeter based on the local weather. The weather stations measure barometric pressure at their elevation and "correct it to sea level" you get this corrected reading over the radio and set it in your altimeter so your pressure-based altitude reading is corrected for local weather variations. Just going out flying for an hour the altimeter setting when returning to the same place might be off by a few millibar.