The definitions are equivalent. Here is the first argument I could come up with. There may well be a simpler one using the full power of the topos assumption; the below works in any well-pointed category with a strict initial object in which every epimorphism is effective.
In a well-pointed topos, if an object $X$ has no points, then we know that $X$ has at most one morphism to any object $Y$. In other words, the unique morphism $q:0\to X$ from the initial object is an epimorphism. The dual property, of $X$ being subterminal, is widely used, but I'm not sure this one has a standard name. "Quot-initial"?
Anyway, in any topos, every epimorphism is effective. The linked nLab article cites Mac Lane-Moerdijk, IV.7.8, for this result. Thus $q$ is the quotient map for some equivalence relation $R\rightarrowtail 0\times 0$ on the initial object $0$. But since toposes have strict initial objects, we find both $0\times 0\cong 0$ and (thus) $R\cong 0$. That is, there is a unique equivalence relation on $0$ and thus $q$ is isomorphic to the identity map in the slice category under $0$.
First, you haven't written down the replacement scheme quite correctly. What was probably intended was $$\forall z_1,...,z_n\forall S(\forall x\in S\exists!y\;\Phi(x,y,z_1,z_2,...,z_n,S)\\\implies\exists R\forall y(y\in R\iff\exists x\in S\;\Phi(x,y,z_1,z_2,...,z_n,S)))$$ (and with a similar modification for the other version).
That aside, the idea you give in your second bullet is a correct argument that the two versions you give are equivalent. In a little more detail, let $z_1,\ldots z_n$ and $S$ be any sets. We need to show $$\forall x\in S\exists!y\;\Phi(x,y,z_1,z_2,...,z_n,S)\implies\exists R\forall y(y\in R\iff\exists x\in S\;\Phi(x,y,z_1,z_2,...,z_n,S))).$$ Per your idea, consider the an instance of the second version of replacement $$\forall z_1,...,z_{n+1}\forall S(\forall x\in S\exists!y\;\Phi(x,y,z_1,z_2,...,z_n, z_{n+1})\implies\exists R\forall y(y\in R\iff\exists x\in S\;\Phi(x,y,z_1,z_2,...,z_n, z_{n+1}))).$$ and instantiate the outer universal quantifiers as $z_i=z_i$ for $i=1,\ldots n,$ $z_{n+1}$ as $S$ and $S$ as $S,$ and then we have exactly what we needed to show.
Best Answer
You'll likely be interested in a discussion about replacement that happened in the category theory mailing list a few years ago. You can find it here. Just grep for "replacement" and you'll get to the right area in the email chain.
In particular, you'll likely be interested in Mike Shulman's question
which Colin McLarty answers as follows (minor mathjax related edits are mine):
The "early published proof" he's referring to is Gerhard Osius's Logical and Set Theoretical Tools in Elementary Topoi. Later in the same email, he mentions his own paper Exploring Categorical Structuralism as a good reference for details on replacement (rather than reflection, as discussed in Osius's paper).
Even though I copied the relevant section here, there are many other interesting tidbits about replacement and other aspects of categorical set theory that you're likely to be interested in scattered throughout that email thread (it's also linked from the nlab page on replacement, which you might also be interested in). It's definitely worth a read!
I hope this helps ^_^