The Elements of Mathematics series by N. Bourbaki intended to construct all of mathematics from a first-order system with equality which he called the theory of sets. (It is not the same as Zermelo-Frankel set theory with the axiom of choice.) In particular, the word set is synonymous with term and every mathematical statement is translatable into the formal language.
Bourbaki defines a formula to be true if it is provable, i.e., a theorem in the theory of sets. Is this definition outdated, in this context?
Contemporary logicians do tend to keep a firmer distinction between "true" (in a model) and "provable" (in a theory). One place this distinction is particularly visible is in the incompleteness theorem, which is often referred to (vaguely) as giving a "true but unprovable" sentence.
A key aspect of Bourbaki's treatment is that it is grounded in essentially pre-1930s mathematical logic. There is a long, somewhat polemical paper "The Ignorance of Bourbaki" (Journal link / Preprint) by A.R.D Mathias, 1992, that examines their approach in more detail.
Of course, Bourbaki's books had other aspects that were also slightly behind the times, such as the minimal treatment of category theory. I think part of this can be attributed to the fact that the original members of the Bourbaki group were simply educated slightly too early to have more contemporary viewpoints on logic or categories.
There is another change since the Bourbaki era, though. In the late 19th and early 20th centuries, a number of logicians and mathematicians had an idea that there is just a need for one, overarching foundational system. Once this foundation is laid, they thought, mathematicians can just go back to their ordinary business, in the same way that I can ignore the foundations beneath my office building when I go to work each day.
A more contemporary viewpoint is that there is a wide collection of foundational theories, each useful for its own purpose. This viewpoint doesn't accept any foundational theory as the final, overarching foundation for mathematics.
However, suppose we did think that all mathematical reasoning was formalizable in some theory $T$ (perhaps because we define mathematical reasoning to be the arguments formalizable in $T$). To know that some formula is true, therefore, we would need to prove it in $T$. So, in that limited sense of "true", it might make some sense to define "true" as "provable in $T$".
In the more contemporary viewpoint, where we are constantly switching between different theories and different models, it makes much less sense to define "true" in that way.