The last few systems I've worked on professionally have been nested PI controllers with some filters on the front or back of them. Just needed the correct framing for the problem (the hard part) and it becomes straightforward to solve.
I'd think that even if you stop applying a delta the second you hit the set-point, if you've built up non-trivial inertia, then you'll overshoot the target fairly significantly?
For the domains I've worked in, with physical components, system response isn't instantaneous, but some level of overshoot is tolerable as long as the system rapidly converges and (this is the important part) cannot end up in a positive feedback scenario. The derivative term tends to amplify noise, which is never what you want.
A little anti-windup, some generous limits around your max extents, and perhaps a gear-shift to switch bandwidths when you approach steady-state, and it's surprising how many problems can be solved with a filter and a PI controller for the closed-loop component of the system. Maybe the actuator is walking you up and down a complicated model space, maybe you've got multiple PIs driving multiple models...but there's a good chance a PI will do a decent job somewhere in the solution.
Use a PID if you want to make more work for yourself.
Specifics will depend on your exact system.
Of course, a "perfectly" tuned PID loop will be "better" - more efficient, faster to reach setpoint, and minimize overshoot. But the effort/engineering time/operation time involved in tuning it, combined with the risks presented by a subtly poorly tuned loop that goes haywire one day when the system paradigm (characteristic) changes...just isn't worth it.
Obviously there are times when you really do need a D term, but I can confidently say that they're not commonly used in HVAC or petrochemical plants. I'm sure there are other domains where the D term is needed, but in my time as a controls engineer I didn't run across them.
A really good way to tell is to consider the relationship between your input and output. PID get's thrown around all the time because a lot of the time you are talking about controlling an accerlation (gas pedal, throttle) to effect a position, so you theoretically need a PID to stand in for a 2nd order differential equation since position and acceleration are two derivatives apart with respect to time. But as you pointed out, you usually can generalize your input and not worry about the D term.
In the video game I'm making, I don't really worry about derivative control, but PID control is mostly there to give the illusion that someone is driving a vehicle rather than it be controlled perfectly by a computer program. I don't want to model out how "skilled" the driver is - all I have to do to give that impression is just give the vehicle better handling.
With a heater,once you get to zero error, any overshoot will reduce back the opposite way because heat is always lost.
Where you do you get the derivative? If you have to differentiate the input, and there is noise, the raw derivative will have huge variations. Then you need a low pass filter before the differentiator. This introduces lag. So you now have both D gain and filter frequency to tune.
See any classical control theory book for the theory on this.
When the rocket had an angle error it needed to point it's thrust to rotate itself back to 0. There is no built in stopping force that stops the rocket rotating when the thrust is not angled. So a D term was needed in order to take into account how fast the rotation was happening. When it was rotating to 0 degrees of error, it needed that d term to not overshoot. With the d term, the thrust would swing in the opposite direction to prevent it overshooting.