Is the reduction correct? Is the following formulation of the reduction correct?  
EDIT: 
Undecidability of an (positive) existential theory $T$ is proved often by reducing an other (positive) existential theory $T'$, which is known to be undecidable,to $T$, i.e., by a mapping from the (positive) existential sentences in the language of $T'$ to the (positive) existential sentences in the language of $T$, $$\phi' \mapsto \phi$$ such that $T'$ proves $\phi'$ iff $T$ proves $\phi$. 
$$$$ 
Is everything correct? Could I improve something at the formulation?
$\phi$ is a formula of $T$, or not? 
 A: I'm not quite sure what you mean by "interpretation" - really, a reduction of $T$ to $T'$ should just be a function $f$ from the sentences in the language of $T$ to the sentences in the language of $T'$, such that $T$ proves $\varphi$ iff $T'$ proves $f(\varphi)$. This is much broader than a (computable) interpretation.
Regardless of exactly how you define interpretation, your second argument is wrong: saying "$T$ is reducible to $T'$" means that $T$ is no more complicated than $T'$. In particular, $T'$ can be extremely complicated and $T$ can be extremely simple. The only way to get a contradiction is to interpret a complicated theory inside a simple theory.
A: It would be best not to use the technical term reduction.  For what you are working with, the following is a reasonable formulation of the strategy. Let $E$ be the set of existential sentences in some language, and let $T$ be the set of sentences of $E$ that are true in some specified structure $M$.  We are interested in showing that there is no decision procedure to determine membership in $T$. 
We usually find a recursive set $E'$ of sentences, and a structure $M'$, with the following properties:
(i) It is known that there is no algorithm for determining whether a sentence of $E'$ is true in $M'$, and
(ii) There is an algorithm which, given as input any sentence $\phi'$ of $E'$, produces a sentence $\phi$ of $E$ such that $\phi'$ is true in $M'$ if and only if $\phi$ is true in $M$.    
For existential (or positive existential) sentences we can be more specific. Ever since the result of Matijasevich, the set $E'$ used has been,  directly or indirectly, the set of existential sentences in the usual language of arithmetic, and the structure $M'$ has been the natural numbers. 
