This idea in play here is due to Rosser and is the main idea behind the [Gödel-Rosser theorem](https://en.wikipedia.org/wiki/Rosser%27s_trick).

Specifically, Rosser proposes to consider the sentence $\rho$ asserting that for every proof of $\rho$ in PA, there is a smaller proof of $\neg\rho$. In your terminology, $\rho$ asserts its own non-strong-provability. Such a sentence can be constructed by the fixed-point lemma. The conclusion is that if PA is consistent, then $\rho$ is independent of PA. It cannot be provable, since then it would have a proof of some specific length, and PA would prove that some smaller number would be a proof of $\neg\rho$, but by consistency none of those numbers can actually code a proof. And it cannot be refutable, since then $\neg\rho$ would have a proof of some specific length, and so PA would have to prove that one of the smaller numbers is a proof of $\rho$, which again can't happen by consistency. 



The forward direction of your biconditional, which you say is "clear", is false if $T$ is inconsistent, since $T$ will prove every $S$, but it will not strongly prove every $S$. Meanwhile, the converse direction is true, since if a statement is strongly provable, it is provable.