Renaming changes the project's display name. It does not create a new project, change its immutable ID, or move its numbers into another workspace. An integration that stores the project ID continues to identify the same project.
Check the ID rather than relying only on a name that another member can edit. If your own application caches a display name, refresh that display separately. A resource genuinely missing after a rename needs investigation, not another project creation. Give support the previous and new names, approximate time, and visible error; never include the integration's full key.