Audit is separate from recount, though the capacity for the former is equivalent to the capacity for the latter. Audits can be done by third parties as a review without having the force of an official recount (for instance, this happened in Florida in 2000 after the official recount was ended.)

If an election method cannot be audited, it cannot be trusted.

It's the same thing. You can call the case where the counting is not official whatever you want, but the same bar exists. "Audit" requests get challenged in court.

Note I said nothing about whether recountability is a good thing. It's clearly a good thing. Hence we should stick with hand counted paper ballots since machine counting introduces trivial ways to game the vote.