Are the definitions of recursively enumerate equivalent?
Your two definitions are essentially the same: a $\Sigma_1^0$ formula is one of the form
$$\exists x Q(n,x),$$
where $Q$ is computable and $n$ is the parameter.

Another difference between the two definitions is that the first only defines r.e. sets, i.e. unary relations, whereas the second defines r.e. relations of arbitrary arity. The two definitions coincide in the case of unary relations.