Even though most of the content of this FF was known to me that last question got me to think. In my very first course on electronics we used the LM358 and found that there was a discontinuity in the negative swing of the output. We found that the TL072 didn't have that 'feature' so we used that instead.
Since I actually didn't know what causes this glitch, even though I had some ideas after Dave's video, I went to investigate. So I bread-boarded Dave's little circuit, put 2Vpp into the input and sure enough there it was, even with the 10k / 100k resistors. (lm358_1.png and lm358_2.png). With the 1k / 10k configuration it is present again, but at a different voltage (lm358_3.png and lm358_4.png).
My first guess was that the glitch has something to do with the output current that the opamps needs to supply. With the 10k/100k configuration this should be about 0.2 mApp. With the 1k / 100k configuration this should be about 2 mApp. Hmm, a first quick glance at the datasheet for Output Current doesn't show a problem yet. The opamp should be perfectly capable of sourcing or sinking 2 mApp / 1mApeak.
Now in the 1k / 10k configuration the glitch sits at about -600mV... Heh! That is a familiar voltage, could it have anything to do with the transistors in the output stage? Let's see where the glitch is in the 10k /100k configuration. Hey, that is about -6.6 V. About 10x (11 actually) as large. That looks to me as if there is indeed something going on with voltage/current and the output stage.
The screen capture in lm358_1 also shows that the glitch is at a slightly different voltage for the negative going swing and the positive going swing. The Datasheet shows the internal schematic output stage. There you can see that there is a small resistor between the top BJT and the bottom BJT. I take it that is used for current limiting. (~0.6 V over the resistor turns on the BJT directly to the left.). Now I also know that a NPN and a PNP of the same 'family' can differ in capabilities. This is seen in the datasheet as that maximum output sourcing current is larger than the maximum output sink current.
All in all my best guess is that the glitch has everything to do with the configuration of the output. I guess that when the input is at 0V the top BJT (darlington) in the output stage is still driving a bit of current through the current limiting resistor. Then, when the input goes positive (thus the output negative) the current going into the output actually goes into the top BJT's before the bottom PNP is turned on. So the output goes negative to about -600mV before the PNP really takes over from the NPN. In the 10k / 100k configuration this is only at -6.6 volts since the current through the current limiting resistor is 10x as small.
Well that's my 2 cents. Hopefully I'm thinking in the right direction.
Attached screen captures:
Ch3: Blue trace: input signal, 1kHz sine, 2Vpp
CH1: Yellow trace: output from LM358
Both: AC coupled, Bandwith limit on, 1M input impedance, normal acquisition mode (not in High Res!)
1) 10k/100k config.
2) 10k/100k config.
3) 1k / 10k config.
4) 1k / 10k config.
edit: added info about screen captures