I know there are a few similar questions on the topic (1,2) , but I still feel they do not fully answer my questions (correct me if I am wrong!). What I am asking is a clarification on the commonly assumed fact that all physical observables - once expressed in terms of other observables - must not depend on the regularization procedure used. As pointed out in (2), it is difficult to define what a regularization procedure is in general, so I am wondering whether any of these apply:
Results have been proven to coincide for the different schemes we know (dimensional regularization, Pauli-Villars, cutoff..etc)
Calculations in different schemes give relations between different physical observables, so it's not always possible to compare them
The only example of regulator independence I have come across is in Matthew Schwartz's "Quantum field theory and the Standard Model", where he shows that a very reasonable class of regulators gives the same results in the calculation of the Casimir force.