There isn't one correct definition of a relativization of a complexity class. Depending on the circumstances there are different definitions which are useful. So we don't have a single way of defining the relativization eve even for a fixed complexity class.
As an example, consider the complexity class $\mathsf{L}$. Using one definition we can show that there is an oracle $A$ s.t. $\mathsf{L}^A=\mathsf{PSpace}^A$. On the other hand, hierarchy theorem relativizes, so $\mathsf{L}^A \neq \mathsf{PSpace}^A$ for all $A$. These results look contradictory at the first sight but they are not. The key point is that they are using different definitions. Both definitions are used in the literature.
Depending on what we want to achieve we have different suitable relativizations. This is similar (though more subtle) to reductions, there are several of them, some results hold w.r.t. one kind of reductions and fails fail w.r.t. another one.
