Make room for the other version
Ask what the other person understood before assuming that you are arguing about the same facts. For a message about how to apologize for giving unclear instructions, use this factual base: state what went wrong, which task or person was affected, and what is already known about the impact.
Correct the record precisely
Correct only what you can support, acknowledge any real impact, and propose a practical way to prevent repetition. Correct the error, name the owner of each next step, and provide a dependable update time. Make the immediate next step clear enough that the recipient can answer without reconstructing the entire history.
Agree on a testable next step
Do not blame workload, hide material risk, or promise a deadline you cannot meet. Use the required manager, HR, security, or compliance route when confidential data or employee safety is involved. Do not recruit an audience, publish private evidence, or turn a misunderstanding into a campaign about character.