For the accepted engineering definition of "damping factor" see for eg. Automatic Control Systems, Kuo; or, Digital Control System Analysis and Design, Phillips and Nagle. Both published by Prentice Hall. The first edition of Phillips and Nagle has a very nice graphical description of how the transient response of a system changes depending on pole location - of course in the z-plan rather than the s-plane seeing as in that instance the analysis is based on on a discrete rather than a continuous analysis (z in that analysis has nothing to do with impedance but comes from the name given for tranforming number sequences to the frequency domain).
Once you review the definition accepted by the engineering community, you will see that damping factor has to do with a system, not merely one component of the system or one set of measurements of a system. The point is that the term, as it has been 'borrowed' is a marketing tool. This is common in the marketing to audiophiles - see the recent thread on the guy who was trying to sell his product to correct for the "doppler effect". The doppler effect has a real scientific definition and by that definition it makes no sense to talk about the doppler effect occurring within an electronic amplifier.
The appropriation of terms in this manner works in marketing like this - from the naive purchasers viewpoint "Oh, the (name the borrowed term) is something I have not considered in my system. It certainly sounds official, and it's "science" so it must be important. Now there is something that I can purchase that will deal with this. If I don't correct for the (name the borrowed term) then my system will not be optimized. I better buy it. Then I will have dealt with another aspect of system degradation." Of course when engineers and scientists debunk the practice, the response shifts to "Oh, I don't know why it works it just does - you can hear it - although sciene and engineering may be able to send man into space they cannot explain the complex world of home audio" or "but these are not simple sine waves - music is complex."
