What is the significance of the standard form of 1st and 2nd order transfer functions?
The order of a transfer function is determined by the highest order of the denominator. This order gives the number of poles and - thus - determines the roll-off characteristics of the transfer function (magnitude) as well as the amount of phase shift for rising frequencies.
The standard form is very important because it allows to find characteristic parameters by visual inspection and/or simple calculations:
Order of the filter, formula for pole frequency, formula for pole-Q.
These parameters are the design entries for designing a filter and can be found for all classical responses in filter tables.
Both forms of the equations can be used, of course. However, the last form is more convenient because you immediately can identify the pole frequency wn. This characteristic frequency is related to the cut-off frequency wc (which normally is given) by a fixed factor which depends on the desired characteristic (Example 1: Butterworth second order, wc=wn; example 2: Chebyshev second order, 0.5dB ripple, wp=1.2313*wc). 


EDIT: I forgot to mention that the pole quality factor (pole Q or Qp) is related to the damping factor ζ (as given in your formulas) by the relation: ζ=1/(2*Qp).  

Summary: The last form (Equ. 6) contains the most important lowpass quantities as parameters (Ao, wp, Qp), which also can be measured (Bode diagram for magnitude and phase). In practice, this form is compared with the form which is derived directly from the circuit - and, thus it is possible to see how all the parameters depend on the particular parts in the circuit.