How can I deduce the damping characteristics formula?

Dear members,

I've got the interest arranging and deducing the formulas defining the damping characteristics of RC lag lead filter.

This is the circuit I've designed:


And this is sort of the arrangements to deduce the damping characteristics formula.


And the problem is that, I'm trying to make the arrangements into the deduced form below:



But this is the arrangement having the deduced form at its end and I could make as much far as possible:


Is there any missing steps or wrong arrangment that I've made in the calculations above?
I would gratefully take your advice and help to improve my knowledge.

Thanks for your interest to my question.

* References:
http://old.iihr.uiowa.edu/~hml/people/kruger/Teaching/ece_55141_2012/Articles/PLLExtract.pdf
http://mars.uta.edu/mae4310/handouts/compensator_realization.pdf
http://www.ece.ucsb.edu/~long/ece594a/PLL_intro_594a_s05.pdf
http://www.ti.com.cn/cn/lit/an/scha003b/scha003b.pdf