TITLE: How to think like a set (or a model) theorist.
QUESTION [6 upvotes]: Kenneth Kunen in his “The Foundations of Mathematics” writes:

‘Set theory is the study of models of ZFC’ (p. 7) 
‘Set theory is the theory of everything’ (p. 14)

With (1) Kunen is pointing to a change in the intended use of the axioms of ZFC: ‘there are two different uses of the word “axioms”: as “statements of faith” and as “definitional axioms”.’ (p. 6).
With (2) he means ‘set theory is all-important. That is

All abstract mathematical concepts are set-theoretic.
All concrete mathematical objects are specific sets.’ (p. 14)

According to (1), to be a set is to be any of the individuals of the universe of a particular model of ZFC, just like being a numeral (standard or not) is being any of the individuals of the universe of a particular model of PA (here I am using Shoenfield’s terminology in “Mathematical Logic”, p. 18).
But, according to (2), models are sets too, as any other objects dealt with in the metatheory.
What's more, models of set theory are defined in terms of relative interpretations of set theory into itself, a syntactical concept. (See Kunen’s “Set Theory. An Introduction to Independence Proofs”, p. 141), which makes the whole thing a bit more confusing.
The view of the axioms of set theory as "definitional axioms" is appealing. And more in regard of (2) since then they pretend to define all that there is. The study of models of set theory has an intrinsic interest, but why reduce the study of set theory to it? Or stated another way, why abandoning the old view?
I would like to know if set theorists do stick to one view or another or shift comfortably between both at need, and the reasons they have to do so.

REPLY [2 votes]: I think the more fundamental question to ask is why set theorists insist that the axioms of set theory be strictly first-order in nature (*).  I claim that you can't really explain the motivations of set theorists until you address this phenomenon.
In light of this near-universal assumption, I think it is fair to say that:

First-order logic is the foundation of set theory (which is in turn the foundation of other things)
The axioms selected for set theory are those which enable as much model theory as possible, without risking inconsistency.

The fact that one can then turn around and use model theory to study set theory itself is a major bonus, but not really a part of the foundations argument.  I've heard some quasi-mystical tales about sets existing in some alternate universe out there, but I feel far more comfortable using "it lets me do model theory and hasn't led to contradictions" as a justification for the axioms.
(*) Excluding, for example, second-order quantification, the logic $L_{\omega_1,\omega}$ of countable conjunctions, or the "exists uncountably many" quantifier.