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).

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.