Is Wikipedia's definition of $\omega$-inconsistency problematic in this way?

Question

I could be wrong, but the definition of $\omega$-inconsistency given at over Wikipedia seems slightly problematic. In particular, Wikipedia claims that $\omega$-inconsistency is a property of a theory $T$, but it seems to me that its actually a property of a pair $(\eta,T)$ where $T$ is a theory and $\eta$ is an interpretation of arithmetic in $T$. The problem is, a theory $T$ can interpret arithmetic in more than one way, I think.

Is this correct, or am I misreading the definition? Here's a copy and paste.

A theory $T$ is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of $T$ so that $T$ is able to prove the basic axioms of the natural numbers under this translation.

A $T$ that interprets arithmetic is $\omega$-inconsistent if, for some property $P$ of natural numbers (defined by a formula in the language of $T$), $T$ proves $P(0)$, $P(1)$, $P(2)$, and so on (that is, for every standard natural number $n$, $T$ proves that $P(n)$ holds), but $T$ also proves that there is some (necessarily nonstandard) natural number $n$ such that $P(n)$ fails.

Answer 1

Just so this question is not marked as unanswered:

Yes, without the interpretation $T$ cannot even express $\exists n \lnot P(n)$. But the phrase "A $T$ that interprets arithmetic" may be understood as "A pair $(T,η)$ of a theory $T$ together with an interpretation $η$". – Hagen von Eitzen Aug 13 at 14:42