I was starting to doubt that the thing I had in my hand really was a capacitor (it was just sitting around on a breadboard) so I swapped it out for a 1 uF electrolytic. Got much the same results as with the other capacitor, with the last circuit above (with the 100 ohm resistor) and a 1 uF capacitor in series with that:
I just don't "get" the shape of the blue line. I obviously need to learn more about capacitors and the way they work in circuits like this.