Estimate the Poles from your RC=Tau (T) values and compare. Your two Types examples, I think do not use the same values.
A larger T for the partial integrator will increase settling time. Choosing filter response with critical damping (Butterworth) or no overshoot (Bessel) or higher BW ( faster response time) as maybe a way to set your goals. These are also tradeoffs with stability, step response error or interference error. This is the same as the methods to compute optimal PID constants Kp, Ki, Kd. Generally ultra-low ESR is a critical factor for ripple with gain which affects filter pole for optimal results. You get to define between what is optimal and what is possible.
This is a not an answer, but a heads-up on how to analyze your filter with different methods for time and frequency domain (Bode or Root Loci)