Electronic – Standard form of 2nd order transfer function (Laplace transform)

What you have initially described is a 2nd order low pass filter then you've made a bit of a complication of things because the bandwidth is \$\omega_n\$ i.e. \$\omega_n\$ is the 3dB point when \$\zeta\$ is not causing the filter to peak i.e. has a value of \$\frac{1}{\sqrt2}\$: -

OK you may be trying to derive a general expression but that doesn't help much visualize the problem so for now I'm assuming \$\zeta\$ = \$\frac{1}{\sqrt2}\$.

Anyway, moving on and ignoring your expression for \$\omega_0\$, the bandwidth of the low-pass filter is from dc to \$\omega_n\$. Then you've modified the 2nd order low-pass filter expression with a 1st order high pass filter (\$s +z\$).

When s is very low, the low pass filter is unaffected other than having a "gain factor" of z. If z is unity then the low pass filter response is unaffected until jw approaches z in magnitude - this then marks a lower 3dB point and the net response climbs with increasing frequency until the original \$\omega_n\$ is reached then, because the low pass filter is a second order, the response starts to fall again.

What you have proposed is a band pass filter with finite gain at dc.

I'm not sure about the bandwidth

If the high pass filter (\$s +z\$) comes into play at significantly lower frequencies than \$\omega_n\$ then the net bandwidth reduces from \$\omega_n\$ to \$\omega_n - z\$.

That's how I see it anyway.

What is the physical meaning of "first" and "second order"? ... How do I know if a system is first or second order?

A 1st order system has one energy storage element and requires just one initial condition to specify the unique solution to the governing differential equation. RC and RL circuits are 1st order systems since each has one energy storage element, a capacitor and inductor respectively.

A 2nd order system has two energy storage elements and requires two initial conditions to specify the unique solution. An RLC circuit is a 2nd order system since it contains a capacitor and an inductor

Where do equations (1) and (4) come from?

Consider the homogeneous case for the 1st order equation:

$$\tau \frac{dy}{dt} + y = 0$$

As is well known, the solution is of the form

$$y_c(t) = y_c(0) \cdot e^{-\frac{t}{\tau}}$$

which gives physical significance to the parameter \$\tau\$ - it is the time constant associated with the system. The larger the time constant \$\tau\$, the longer transients take to decay.

For the 2nd order system, the homogeneous equation is

$$\tau^2\frac{d^2y}{dt^2} + 2\tau \zeta \frac{dy}{dt} + y = 0$$

Assuming the solutions are of the form \$e^{st}\$, the associated characteristic equation is thus

$$\tau^2s^2 + 2\tau\zeta s + 1 = 0 $$

which has two solutions

$$s = \frac{-\zeta \pm\sqrt{\zeta^2 -1}}{\tau}$$

which gives physical meaning to the damping constant \$\zeta\$ associated with the system.

The transient solutions are, when \$\zeta > 1\$ (overdamped), of the form

$$y_c(t) = Ae^{\frac{-\zeta +\sqrt{\zeta^2 -1}}{\tau}t} + Be^{\frac{-\zeta -\sqrt{\zeta^2 -1}}{\tau}t} $$

when \$\zeta = 1\$ (critically damped), the solutions are of the form

$$y_c(t) = \left(A + Bt\right)e^{-\frac{\zeta}{\tau}t} $$

and when \$\zeta < 1\$ (underdamped), the solutions are of the form

$$y_c(t) = e^{-\frac{\zeta}{\tau}t}\left(A\cos \left(t\sqrt{1 - \zeta^2}\right) + B\sin \left(t\sqrt{1 - \zeta^2}\right) \right)$$

When given a first order system, why is sometimes equation (2) given, and sometimes equation (3) as the transfer function for this system?

Different disciplines have different conventions and standard forms. Equation (2) looks to me like control theory standard while equation (3) looks like signal processing standard.

Standard forms evolve to fit the needs of a discipline. Further, if a particularly influential person or group develops and uses a particular convention, that convention often becomes the standard. It might be educational to peruse older textbooks and journals to get a sense of how notation and standard forms evolve.