Wikipedia hints at the definition of RE-complete. A language $A$ is RE-complete if for every r.e. language $L$ there is a computable reduction $f$ such that $x \in L$ iff $f(x) \in A$. There are two features of this definition:
We consider many-one reductions. This allows us to distinguish RE-complete problems from coRE-complete problems.
We allow any computable reduction. This is the strongest canonical class of reductions that still makes sense, since R is a proper subset of RE.
Both of these features are shared by the definition of NP-completeness.
As for R-completeness, I am not aware of any such definition, and I don't expect one. Indeed, the class of reductions allowed has to be smaller than all computable functions, since otherwise the definition trivializes (all nontrivial computable languages would be R-complete). Suppose for definiteness that we consider reductions running in time $t_1(n)$, and let $L$ be some R-complete problem with respect to such many-one reductions. Then $L$ runs in time $t_2(n)$, and so we would get that all languages in $R$ can be computed in time $t_1(n) + t_2(t_1(n))$, which clearly isn't the case. The same kind of argument rules out oracle reductions and space-bounded reductions.