The important point for me is that the equivalence for Cauchy sequences are part of the definition of real numbers as Cauchy sequences. This ought to imply that one has to be able to decide equivalence of two sequences for the definition to make sense. For Dedekind cuts, the crucial aspect is being able to define the set and that is something that can be called into question. But if that is done, it is just a computational question in comparing two Dedekind cuts, not a definitional one.