Electronic – PID controller – should terms be completely independant

You're getting befuddled by all of the fanciness of the Z-transform. The two approaches are fundamentally the same - the PID without PHD approach just has fewer subscripts. They perform the same basic function and use the same basic math.

The only major difference between the two that I can see is that the PID without PHD doesn't take sampling time into account. For doing anything that might be unstable, sampling time is a very important consideration. The benefit of the Z-transform approach in this case is that you can't use it without taking sampling time into account - it forces you to show your work and helps you design a more stable system.

It also looks like the case study you found implementing the Z-transform approach was designed to be highly deterministic. This explains their use of FPGAs - the calculations will always take the same amount of time. The PID without PHD implementation is decidedly not deterministic. The use of doubles as variables instead of a fixed-point implementation is sure to cause non-deterministic behavior on any microcontroller without a floating-point unit (and probably on uCs with an FPU as well). The case study is working on a whole different level of complexity compared to the PID w/o PHD approach.

So fundamentally the math and control approach is the same, but the case study/Z-transform approach is more rigorous and theoretically grounded. The PID w/o PHD approach will only work for very simple, non time-critical system that are relatively stable.

It's not a complete answer but I hope that it could be of some help.

You could rewrite the first system as

$$ \begin{cases} P(n) = K_P E(n) \\ I(n) = I(n-1) + \frac{K_P}{T_I} E(n) \Delta t \\ D(n) = K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \end{cases} $$

Where \$E(n) = G(n) - target(n)\$ and \$\Delta t\$ is your sampling interval. Note that \$T_D\$ and \$T_I\$ are not defined as gains. \$K_I = \frac{K_P}{T_I}\$ and \$K_D = K_P T_I\$ are respectively the integral gain and the derivative gain.

Now you can rewrite the system as a single function of the error.

$$ PID(n) = P(n) + I(n) + D(n) $$

$$ I(n-1) = PID(n-1) - P(n-1) - D(n-1) \\ = PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} $$

$$ PID(n) = K_P E(n) + PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} + \frac{K_P}{T_I} E(n) \Delta t + K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \\ = PID(n-1) + K_P \left(\left(1 + \frac{\Delta t}{T_I} + \frac{T_D}{\Delta t} \right)E(n) - \left(1 + 2\frac{T_D}{\Delta t} \right)E(n-1) + \frac{T_D}{\Delta t} E(n-2) \right) $$

The second one is a bit more complex to rewrite as a single equation but you can do it in a similar way. The result should be

$$ R(n) = K_1 R(n-1) - (\gamma K_0 + K_2) R(n-2) + (1+\gamma) (PID(n) - K_1 PID(n-1) + K_2 PID(n-2)) $$

Now you only need to substitute the equation of the PID in order to obtain the equation of the regulator as function of the error.