If you think the British would have returned New Orleans based on that technicality I have to disagree.
All British territory below the candian border was ceded and all brit troops south of the border were removed.
Terms of the treaty were significantly influenced by the Duke of Wellington who stated pretty bluntly that concession of any territory wasn’t feasible despite military successes they had achieved in Washington, Maine, etc.
Treaty returned to the status quo including return of parts of Maine they had already won and wanted to incorporate into Canada as well as large tracts of the Midwest. Treaty was not finalized by Congress until Feb 16 more than a month after the Battle of NO on Jan 8th and only 4 days after the final battle at Fort Bowery. The Senate knew of NO by that time and would not have ratified the treaty with NO remaining in the Brits hands. The Brits wanted the war done and over - finances, public opinion, and France was hinting at rejoining. Had the attempted to retain NO we would have continued till they were out.
.................................... Well look at this, Eric Johnson, the reporter of Seattle is Dying is doing another blockbuster on KOMO 4 now. Butt kicking! Horrible situation. In your face truth of bad 'compassion' decisions.