If I change the live copy page title by going to page properties, then the relationship between source and the live copy is maintained. But if I am changing the node name from crxde, then I do see the "live copy does not exist" status in the overview.
Rename should be done from the blueprint. In other words, before performing the rollout, you need to update the blueprint page name. So, that, after the rollout, livecopy name is updated.
You can also, look into creating custom rollout configs that renames the livecopy and still maintains the relationship but the way you are doing it is not the recommended way and will cause the blueprint/livecopy link to break.