Leading with "an idempotency key makes a retry safe" is a good choice, because it is the one most engineers would defend without thinking.
The key only makes a retry safe if the effect and the key are committed atomically. If the send happens and the record of the key does not, you get the worst possible state: the money moved and the system believes it did not.
That is structurally the same bug as at-least-once delivery with a non-idempotent side effect, which is why it turns up far outside payments. The difference is only that everywhere else the duplicate is cheap. Looking forward to the other nine, and the penny that switched off the balance check is a very good opening.