$1$-consistency of $PA$ is not provable in $PA$+{all true $\Pi_1$ sentences} (see this article). Consistency statements are all $\Pi_1$ sentences, so it is not equivalent to any $\Pi_1$ sentence.

$1$-consistency of $PA$ is a true $\Pi_3$ sentence which is not provable in $PA$+{all true $\Pi_1$ sentences} (see this article). Simple (iterated) consistency statements (as you mentioned above) are all (true) $\Pi_1$ sentences, so it is not equivalent to any $\Pi_1$ sentence.