Thanks for this detailed reply. To answer your questions in order.
No one has reported back to me that they have received the OOO messages on old emails since the original problem occurred and I deleted the rule so I think they have stopped but cannot be absolutely sure.
The account in question is Gmail and iCloud was not activated for mail on this or any of my devices (iPhone and 2 iPads).
I cannot remember exactly what I did but if there was an apply button at the end of the Rule setting Process I would have clicked it assuming I was applying the rule. How is someone supposed to know that they are applying the rule retrospectively? I realise the system doesn’t know your rule is an OOO auto reply, but I cannot imagine why you would want to apply any new rule retrospectively?
The rule has been deleted, the Mail app closed, the mail account disabled on the device (MacBook Pro) and the device rebooted.
The only other rule that exists is one that I think Apple instal about Apple News? I certainly didn’t create it.
I need to feel confident that I can reenable the account on the laptop and not have the problem occur again because of some time lag in the rule or some way it is embedded that I haven’t removed. For example would it be a good idea to delete the account from that device and then reinstall it? Would that help to be sure that it is ‘clean’?
Thanks for your help.