From this answer,
(...) "not P" should be interpreted as the assertion "P implies a contradiction".
Is this the (only/widespread/mainly) accepted definition of the negation of a statement? For me, it definitely makes sense, as a false statement ultimately leads to the negation of an axiom.
Well, $P$ is logically equivalent to $\neg P\Rightarrow\text{false}$ and is the basis of the proof of contradictio. This coincides with your observation.