I can see advantages and disadvantages to the various monitoring methods, but I think if wort stability is the goal, then ambient stability is the answer. If those temps go up and down in, say, a four degree range, they are not going to alter wort temps much at all, and you'll never see as much as a two degree swing in wort temp caused by those ambient changes.
I would disagree that ambient air is the way to go. Before and after fermentation, that would work OK. However, during vigorous fermentation, the temperature will rise rapidly in the wort. The freezer will turn on and cool the AIR to your setpoint. However, the mass of air that has cooled to the setpoint (and caused the controller to shutoff the freezer) will not be sufficient for actually controlling the actual wort temperature. It takes a lot less energy to heat air than water. You will likely get either the freezer kicking on and off frequently (if you have a shorter ASD), or the wort will be anywhere from 3-10 degrees higher than the ambient air if you have a longer ASD. You could set the controller low, but you may then (1) cool the wort significantly lower during the lag phase, so fermentation will have a longer delay, (2) have the chance of differing levels of fermentation vigor batch-to-batch (lager --> saison, also different strains in same temp regimen), resulting in different heat production and different offsets required to maintain your setpoint (3-10F I would say is a realistic range, so picking 4F may not be sufficient for the most vigorous fermentations), and (3) once fermentation dies down, temperature will drop back to that point. You could babysit the beer and adjust as it goes, but most don't have the time for that.
Also, taping to a glass of water is not ideal either. The water will take much longer to warm from the heat given off from the fermenting carboy than air will, and your actual wort temperature will be higher than setpoint during active fermentation.
You should go back and reread the posts authored by jbaysurfer, BigFloyd, and CryoEng (thermal physicist apparently) in the second link. They cut to the heart of the matter. Cooling the air by 10F will not significantly affect the wort temperature unless the temperature differential is held for a considerable time. I would argue that while you may control the ambient air temperature better, during active fermentation you would be allowing your wort to heat up (maybe significantly) over the setpoint and have your freezer turning on/off continuously to try and 'catch up'.
Graphs from Reddit Link
The graphs shown in the link above give a good idea of how effective the probe taped to the side can be. Wort was around 1-1.5F higher in the fermenter compared to on the side. This lines up with other folks claims that the probe taped to the side is within 1-2F of the actual bulk wort temperature. Unfortunately, the homebrewer did not also measure ambient air.
Here a homebrewer has measured the thermowell and ambient air temperature:
Thermowell vs Ambient
You can see that to maintain the wort at 64F, the air in the freezer was actually 56-60F to maintain the setpoint. This shows how little the wort temperature actually changes with short-term air temperature fluctuations. Of course, this brewer used a more intricate control scheme than just the thermowell, but there is a good point to be had anyhow. I would imagine the fluctuations in wort temperature would be around the 1-2F range, rather than 0.5F or less when using a more complex controller. This brewer notes that controlling using air temperature can result in better control, but only if you babysit the fermentation. Most of us work and don't have time to do that.