I like your paraphrase but I think it muddies the water. We really want to prove that axioms of a system cannot be contradicted if the system is inconsistent.
To be even more precise, the system or theory T proves that there is no number n which provides a proof of contradiction for the axioms of T.
So its the axioms of the systems which all formal systems assume and hold to be true. Basically its impossible to prove that axioms are true with the same system.
In short, no formal system can define its own consistency because in order for its axioms to be true, the system must be inconsistent.
To be even more precise, the system or theory T proves that there is no number n which provides a proof of contradiction for the axioms of T.
So its the axioms of the systems which all formal systems assume and hold to be true. Basically its impossible to prove that axioms are true with the same system.
In short, no formal system can define its own consistency because in order for its axioms to be true, the system must be inconsistent.