Soundness multiple definitions I had asked a question earlier about multiple definitions for soundness of a formal system (with a standard first-order logic). Following were the two definitions

A theory $T$ is sound iff its axioms are true (on the interpretation built
into T’s language), and its proof system is truth-preserving, so all its theorems
are true.

This definition to me means what one can prove syntactically using the formal system $T$, is
indeed true for the built-in interpretation (assuming I have a way "independent" of $T$ to check what is "true").
The second definition (based on the clarification from the comment on my previous question) for soundness says

$T \vdash \phi$  only if $T \models \phi$
where $T \models \phi$ means any way of (re) interpreting the non-logical vocabulary
that makes all the theorems (not only axioms) of $T$ true makes $\phi$ true.

But this raises another question for me. Here a formal system $T$ is sound, if whatever it can
prove syntactically is true in all the interpretations that make all theorems of $T$ true.
But all the interpretations that make all theorems of $T$ true, will surely make any sentence true that can be syntactically proven by $T$. Won't this make any $T$ sound ?
 A: 
But all the interpretations that make all theorems of $T$ true, will surely make any sentence true that can be syntactically proven by $T$.

Yes... if the formal system is sound!
It's understandable that you should expect that property to hold, because it is a very sensible to property to have, but it is not something that is just true by accident. The rules of the proof system you have in mind were carefully chosen to satisfy this property. But if you stuck an extra rule in that said "from $A\lor B,$ infer $A$," you would find that this property no longer held, i.e. the formal system would no longer be sound.
Alternatively and less trivially, if you changed the semantics, say, from classical to intuitionistic, while keeping the proofs the same, the resulting system would no longer be sound… this could be rectified by changing the proof system to an intuitionistic one.
It's important to appreciate that semantics and deductive systems are in principle things that could be chosen independently of one another, even if we generally package together systems that have these nice properties in how they interrelate.
(I would also advise you to look into some modern treatment of logic. You are conflating things that are generally kept separate. For example a language doesn't have a "built-in" interpretation. And usually the nonlogical axioms (i.e. the "theory") are not considered part of the logical system, but something that is variable. This looser coupling has a lot of advantages and will clarify the issues in question.)
