“Slow” sounding bass doesn’t usually (if ever) have anything to do with actual speed. It gets one noted because of an and/or situation between stored energy & feedback - feedback that can sometimes be a partial result from room reinforcement or enclosure style.
On the other hand “fast” bass can come from elongated signal from transformer output tube amps, tubes themselves, and classD’s output filters. At least those are common.
Neither are “right” exactly. But people clearly enjoy different types. I do like well defined bass, and am not really a fan of ported speakers. But I have heard some that still sound real good.
Pistonic, it means nothing more than the driver moves like a piston. Nothing more. You could interchangeably use “dynamic” (probably the dumbest term for a driver). It just so happens that some like to use it in marketing to differentiate themselves with their stiffer cones. It’s not like anyone purpsoely tries to make flexy cones... but there can be trade offs to stiffness shape being dominate.