Approximately how many types of stress are known/defined? Ex. Piola stress, Cauchy stress ...
Your examples aren't really "types" of stress, but different ways to measure stress. The real world structure just responds the way it responds - it doesn't worry about what sort of math you used when you tried to model its behaviour.
https://en.wikipedia.org/wiki/Stress_measures lists a handful different stress measures that have acquired "names". Cauchy stress ("true stress") and Second Piola-Kirchhoff stress are probably the most commonly used.
It doesn't really make sense to talk about a measure of stress in isolation, without also considering how to measure strain, and how to model the constitutive equations that give the relationship between stress and strain.
For example 2-PK stress works nicely with Green strain, which is a convenient way to define the behavior of a body which has "small" elastic deformations superimposed on arbitrary large rigid body motions - in particular, large rigid body rotations, where approximations like $\sin \theta \approx \theta$ are not appropriate.