Understanding the Definition of the Axiom Schema of Specification Consider the Axiom Schema of Separation:

If $P$ is a property (with paramter $p$), then for any $X$ and $p$
  there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those
  $u \in X$ that have property $P$.

Now consider two potential interpretations of this axiom schema:

(1) If $X$ is a set, and $P$ is any arbitrary property, then we can
  specify a subset of $X$ in which all of the members of this subset
  satisfy $P$.
(2) If $X$ is a set, and $P$ is any arbitrary property that can be
  expressed in terms of a finite number of expressions involving only
  the relations of $\in$ and $=$ and the logical connectives, then we
  can specify a subset of $X$ in which all of the members of this subset
  satisfy $P$ given the constraints in bold.

My question is am I correct in assuming that (2) is the proper way to understand this axiom schema, and that (1) and (2) are strictly distinct from each other?
 A: The magnificent beauty of the axioms of $\sf ZF$ is that they allow us, with only $\in$ to express so much.
When we say an arbitrary property, we mean one that can be expressed in the language of set theory. Otherwise it will be impossible to write the axiom relevant to that property in the language of set theory, which is the language of $\sf ZF$.
You seem to forget that there are quantifiers to be used in the formulas. Not everything is boolean combinations of atomic formulas and their negations. No, we make heavy use of quantification here. For example $\subseteq$ can be defined as $x\subseteq y\iff\forall z(z\in x\rightarrow z\in y)$, and we can define when a set is transitive, $\forall y(y\in x\rightarrow\forall u(u\in y\rightarrow u\in x))$, or in shorter form, $\forall y(y\in x\rightarrow y\subseteq x)$.
We can define $x\cup y$, and $x\cap y$, and more and more. Some of the formulas require us to rely on the axioms in order to prove their correctness, but that's fine. We are allowed to do that. But we can sit and write formulas which quickly become more and more complicated and those express a lot. A lot more than just $x\in y$ or $x=y$.
