But what I am most interested in is to learn how to interpret cone break-up and if it can be found in various objective measurements such as frequency response, cumulative spectral decay, impedance etc.
You've now seen examples of woofer break up in frequency response and cumulative spectral decay (waterfall) plots.
You can also see evidence of woofer break up in impedance plots. In
this example, you see a 7" woofer with a kevlar/paper cone. Scroll down until you can see the FR and Impedance plots. This is the woofer's raw unfiltered response, not in any specified cabinet, probably mounted on a large baffle.
Look first at the FR curve (top). There are some small irregular peaks just above 1 kHz, and several larger peaks above 2.5 kHz. At first glance, I would think the ~1 kHz peak isn't too bad, but I'd definitely crossover this woofer low enough to avoid the 2.5 kHz and higher peaks.
Now look at the impedance curve (lower figure). (There are two curves both in black. One is impedance magnitude vs. frequency. It has the large peak below 50 Hz indicates the Fs frequency of the woofer - ignore that peak, its not a problem. The other curve is impedance phase vs. frequency.) Above the large Fs peak, the impedance magnitude curve should gradually rise smoothly. Notice that just above 1 kHz there is some small unevenness in the gradually rising impedance. Most important is that the FR and the impedance curves both show something suspicious at the same frequency, just above 1 kHz. This is a good indication of the beginning of break up or a funky resonance developing between the kevlar/paper cone and the rubber surround. Notice also that this impedance bump is larger than the ones around 3 kHz where the FR curve shows worse looking break up peaks. This kind of impedance curve combined with the FR curve is enough for me to say, drop this woofer and look for something else.
Also, if cone break-up is not dealt with properly its impact on overall perceived sound quality when doing listening tests.
In my opinion, if you can hear woofer break up, it is the worst sounding noise an otherwise undamaged speaker can make. It means the crossover is poorly designed. Start over.
Most people blame woofer break up noise on the tweeter only because the noise is high frequency.