Currently if I move or rename a published page, the old page will be deleted on the publisher side automatically. I even try uncheck the republish option and the old ones still go disappear! This seems to be a default behaviour, is there any way I can keep the old pages on publisher side? Much appreciated!
As per Adobe, If the page was already activated, moving the page will automatically deactivate it. By default, it will be reactivated when the move is complete, but this can changed by unchecking the Republish field for the page in the Move window.
1. If a page is already activated, moving or renaming it will deactivate it anyhow which will remove the page reference from publish instance.
2. Now if the Republish checkbox is checked, then the updated (moved or renamed) page will be moved to publish.
3. If it's not checked, the new page will not be moved to publish but at this point #1 is already completed. So we will not be getting the previous page back here.
It is not recommended to modify this feature as it's the OOTB functionality. But if still you want to try out something you can check for Pre-Processor. Pre-processors are invoked just before the activation/deactivation. This will run for all type of activation and deactivation and is not limited to page move action.
How do you want to make changes to the old page residing on publish, if you don't have it on author anymore? How do you want to delete it? Are you authors even aware that there is an orphan page on publish which they don't see on author?
Making things simple and not changing the out-of-the-box AEM behavior, for your requirement, I would manually copy the page (CTRL+C), and then paste (CTRL+V) and rename the page (using the move tool, using the keyboard button "M").
Also, check with your SEO team member if having duplicated web pages on your AEM website is good practice or not.