Are the definitions of recursively enumerate equivalent? There are a couple of definitions of recursively enumerable, for example in Judah: $A \subset \mathbb{N}$ is called r.e. if there exist a $\Sigma^0_1$ formula $\varphi(x)$ such that

$$A:=\{n \in \mathbb{N}: \mathbb{N} \vDash \varphi(n)\}$$

In Shoenfield: A predicate $P$ is r.e. if there is a recursive predicate $Q$ such that

$$P(\bar{a}) \leftrightarrow \exists x Q(\bar{a}, x)$$ for all $\bar{a} \in P$. And in here there is another one A new definition of recursively enumerable set?.

My question is: Are the definitions of recursively enumerate equivalent? Why?
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.