Hi, edited shared project "name" doesn't reflected/updated at my collaborator's end.
This is the correct behaviour. This allows you customize your projects even if they are shared. Also, the level of the project is not shared as well for the same reason.
We'll look into changing this in the future. You can also add a suggestion in the Votebox for this https://todoist.com/Vote/.
We'd find it very helpful as well if we could use that feature and be able to change/update project names so it is reflected in our collaborator's todoist desktop.