Do the everyday mathematician and the model theorist mean the same thing by "truth"? In Terrence Tao's book Analysis I, the axioms of ZFC are considered to be true statements, and every other true statement in the book  is proved from these axioms. Model theory is not mentioned.
This approach to truth is not peculiar to Tao's book or to its subject.
As far as I can tell, this is the standard approach for books and research papers outside of mathematical logic and set theory.
I have two questions:


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*When the everyday mathematician says that a certain statement is true, does the model theorist agree? Why or why not?

*Why does the everyday mathematician need the model-theoretic definition of truth, if only provability is ever used?


 A: It's important to distinguish truth in a structure (e.g. "$\varphi$ is true in $M$") from unqualified truth (e.g. "$P$ is true").
The former expresses a mathematical relationship, precisely defined by the model-theoretic definition of truth, between a sentence $\varphi$ of a logic and a structure $M$, which is a set possibly with extra functions and relations defined on it. It's important here that both $\varphi$ and $M$ are mathematical objects.
The latter is a linguistic construction that we sometimes use at the meta-level, i.e., when we talk about mathematics. What we mean when we say "$P$ is true" is really a question of philosophy of mathematics.
Model theorists, of course, may use both senses of "truth" and mean different things by them. I think this answers your two questions:

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*When the everyday mathematician says that a certain statement is true, does the model theorist agree?

This depends on whether their views on philosophy of mathematics align. No one has mentioned truth in a structure in this scenario, so there is nothing special about model theorists here - the same holds between two "everyday mathematicians" or between two logicians.


*Why does the everyday mathematician need the model-theoretic definition of truth, if only provability is ever used?

They don't. The model-theoretic definition applies in a very specific context where we are considering logical sentences as mathematical objects. That is, it is only relevant when we are doing mathematical logic.
But! That's not to say that "provability is all that's used". An earlier version of your question asked whether everyday mathematicians identify truth with provability. I would say that most of them do not. To explain a bit more, I'll sketch two common views on philosophy of mathematics (there are many more, and these two views also have many variations). Let's contrast the meanings of the assertions "$P$", "$P$ is true", and "$P$ is provable" (implicitly: from a background foundational system, say ZFC), for people holding these views.
A Platonistic View: There is a canonical "real" universe of mathematical objects. To address a comment in an earlier version of your question: the mathematical universe is not $V_\kappa$ for some inaccessible cardinal $\kappa$, because $\kappa\notin V_\kappa$, and the universe contains all mathematical objects.
For the platonist, asserting "$P$" is the same as asserting "$P$ is true", and it's about what holds in the real mathematical universe. It is different from asserting "$P$ is provable". For example, there is a fact of the matter about whether there is a bijection between the real numbers and the first uncountable ordinal. That is, either the Continuum Hypothesis ($CH$) is true, or $\lnot CH$ is true, but we know that neither is provable from ZFC.
Unfortunately, proof is the only means for us mere mortals to be sure of what is the truth. So the platonist is unlikely to assert "$P$ is true" unless they have a proof of $P$ (or they are feeling reckless or provocative). For example, the platonist is likely to say "I believe ZFC is consistent", rather than "ZFC is consistent", because this statement is not provable in ZFC.
A Formalist View: There is no such thing as mathematical truth, because ideal mathematical objects do not actually exist in the world. For the formalist, asserting "$P$" is the same as asserting "$P$ is provable". It is not the same as asserting "$P$ is true", because "$P$ is true" has no meaning.
For the formalist, the fact that neither $CH$ nor $\lnot CH$ is provable settles the matter. But they might be interested in conditional statements like "such and such additional axiom of set theory implies $CH$", and they do not need to be convinced that the additional axiom is "true".
Also, the formalist might say something like "I believe that it's true that ZFC is consistent" and mean this as a statement about our physical world in the sense that it would be impossible to actually write down a proof of a contradiction from the axioms of ZFC.
As I said above, there are a wide variety of views on philosophy of mathematics and the nature of truth. I'm not suggesting that all platonists or all formalists agree with everything I wrote in my caricatures of these views.
