Is a Type 2a PID(ish) controller just an example of an augmenting integrator?
As you probably know, the op-amp circuit is working inside a feedback system thus the DC operating point of the op-amp is sustained. 

When the output signal is changing rapidly the integrator (without the series feedback resistor) presents a very small feedback signal - basically it's an integrator but, with the resistor in place the dominant impedance under these circumstances is that resistor.

Remember I'm talking about a scenario where the "process" output is changing fairly quickly and therefore the cap looks like a short and the gain of the op-amp is -R0/R1 (ref your bottom circuit) - this is good-old-fashioned proportional control and it drives the "system" into "lock" much quicker than if a pure integrator is used. Once close to "lock" the capacitor starts to dominate and the basic gain of the op-amp circuit rises from the baseline of -R0/R1 to that of an integrator.

I had to design a self-tracking FM demodulator once and the local oscillator had to rapidly lock-on to the carrier frequency. I could have used a pure integrator but I got significantly faster lock-in times with series R+C.

Anyway that's a non-mathematical explanation - it's the P and I of a PID controller and, if you put a series RC across R1 then it's a type of PID controller.