z-Transform Methods: Definition vs. Rectangular Rule or Tustin's Rule
\$\small z^{-1}\$ is the delay operator in the z-domain, ie, multiplying the z transform of a signal by \$\small z^{-1}\$ delays the signal by one time increment. For example, if \$\small X(z)\$ is the z-transform of the unit step sequence: 

$$\small X(z)= z^{0}+z^{-1}+z^{-2}+z^{-3} ...$$ 

then 
$$\small z^{-1}X(z)= \:z^{-1}+z^{-2}+z^{-3} ...$$
which is clearly the delayed unit step (since \$\small z^0=1\$)

Similarly, multiplying a the laplace transform of the unit step by \$\small e^{-sT}\$ delays the step by \$\small T\: sec\$ in the time domain.
Hence the relationship:
$$\small z^{-1} \leftrightarrow e^{-sT}\:\: or\:\:z\leftrightarrow e^{sT}$$

This relationship is useful for pole-zero mapping between the s-, and z- domains, hence can be used for obtaining a digital controller from a continuous, Laplace prototype.

The bilinear transform ('Tustin' or many other names) is an algebraic method of doing, essentially, the same thing, but with generally better perfomance compared with pole-zero mapping, impulse invariant, step invariant, etc

It can be shown quite simply that the 1st order z-transfer function of integration (i.e. trapezoidal integration) is $$\small \frac{T}{2}\frac{(z+1)}{(z-1)}$$ and this corresponds to the Laplace integration operator: \$\large \frac{1}{s} \$

This leads to the very convenient algebraic substitution (i.e. the bilinear transform):$$\small s\leftrightarrow \frac{2}{T}\frac{(z-1)}{(z+1)}$$

Note that this transform preserves DC gain.

For example,say we require digital equivalent of the low-pass filter: $$\small G(s)= \frac{1}{3+s} $$

given a sampling increment, \$\small T=0.1\:sec\$

$$\small G(s)= \frac{Y(s)}{X(s)}=\frac{1}{3+s}\rightarrow G^*(z)=\frac{Y(z)}{X(z)}= \frac{1}{3+\frac{20(z-1)}{(z+1)}}=\frac{z+1}{23z-17} =\frac{0.043+0.043z^{-1}}{1-0.739z^{-1}}$$

The digital filter is thus implemented by the difference equation:
$$ \small y(k)=0.043\left(x(k)+x(k-1)\right)+0.739y(k-1)$$