Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them?
The three definitions that I'm listing all have that $a \geq b$.
Definition 1: $$ \varepsilon = \frac{a-b}{a} = 1 - \frac{b}{a}. $$ This definition comes from my "Dictionary of Physics and Mathematics" from McGraw-Hill.
Definition 2: $$ \varepsilon = \sqrt{\frac{a^2-b^2}{a^2}} = \sqrt{1 - \frac{b^2}{a^2}}. $$ This definition comes from http://mathworld.wolfram.com/Ellipticity.html. As a side note, the Wolfram page on ellipticity also defines something called flattening that is equivalent to my definition 1.
Definition 3: $$ \varepsilon = \frac{a^2-b^2}{a^2+b^2}. $$
I've done a light search for information on definition three, but I have not turned up anything in the literature.