Armadillo: First about your LCR meter.
The 19,38 uF is Cs, series capacitance, D=0,182 so Xc=8,2 ohm Rs=1,49 ohm Z=8,334
The 6,431 uF is Cp, parallel capacitance, re-calculating with D=1,45 gives Cs=19,95 uF, Xc=7,977 Rs=11,56
So your meter is correct, but I hope the meter tells you it is measuring Cp instead of Cs. Cs = Cp only if D is very low.
I do not know what you mean with a squarewave. The meters I measured to test Capacitance after repair used DC to charge the cap and measuring the time. See several service manuals from for instance Fluke. You can measure ESR with a squarewave.
Then the multimeter test, here I'm surprised because I never tested it like that on a DMM. (I did for bridges to calibrate D) My theory is based on experience with caps with real bad ESR and not simulated ESR. So I now tested both situations. And indeed with 39 ohm simulated ESR both my DMM's see almost no difference in capacitance. I can not explaine that. But a real bad cap does what I wrote. (i have tested that many times in the past. see for instance the picture below from a test on caps together with a TV repair man the results are from his caps and his meters checked by me.)
If I find time I will do more measurements and make a video and blog page about that.
My short test like yours and Joes:
100uF panasonic FM
IET DE-5000 93,15 uF, D=0,057 at 1 kHz 95,61 uF and D=0,019 at 100Hz,
Agilent U1252A DMM 101,5 with Rs=39 ohm and 101,4 without
Brymen BM869s DMM, 92,2 using the 39 ohm and 98,9 without
I measured some more but results are about the same:
IET: 10,087uF D=0,112 at 1 kHz
IET 10,084 uF D=2,54 at 1 kHz with 39 ohm
Agilent 11,03 uF alone versus 11,00 with 39 ohm
Brymen 10,85 uF alone versus 10,77 with 39 ohm
That is all nice but these are good caps with a series resistor. I always keep a small collection bad caps for testing. I have a lot of LCR meters, bridges, leakage testers and RCL standards. My pride and joy a GR-1620 that comes out GR's own standards lab. For my daily work (repair of measurment gear and calibration gear) I use a IET DE-5000.
So I took a real bad 470 uF, the IET: Cs 276 uF with a D=1,092 at 1kHz Xc=0,576j Rs= 0,6296 ohm. So no extreme ESR if you look at ohms (but a D of 1 is bad)
The Brymen: 457,3 uF
So if you would measure this cap with an ESR meter and C with a DMM it would be a perfect cap
And a bad 100 uF, IET: 76,45 D=1,281 at 1 kHz, Xc=2,082 j, Rs=2,081 D is bad, capacitance is just outside specs (20%)
Brymen: 108,3 uF
I did not note the Agilent because it was in the same ballpark as the Brymen
So now the 1000 dollar question, why do they ignore a real series resistor upto a certain level like Joe showed but go very wrong with a real bad cap (like I told before)
I did not test more because an industrial calibrator is spread out in pcb's all over my bench for testing so no room.