The ESR meter will measure the 19mOhm (Z= SQRT(X2 + R2).
Sorry, but I don't understand!
Z is impedance (in Ohms);
R is (pure ohmic) resistance - i.e. ESR what we want to measure, right?
X is reactance (in Ohms) - i.e. complex AC resistance that depends on frequency, true?
I can't sketch this, but R and X are two vectors on complex plane under 90° angle, true?
So, if we have, say, 100uF capacitor, its reactance will be 0,0159 Ohm (or 16mOhm as you said).
Impedance will be 18,78 mOhm (if we assume that ESR=10mOhm), again as you said.
BUT IMO ESR meter WILL NOT (or better to say: SHOULD NOT) measure 19 mOhm! Isn't the main feature of every ESR-meter to separate X and R and to show on its display only value of R, i.e. ESR? It should show 10mOhm.
Sorry if I said something stupid, I repeat: I'm not some expert!