If I rename a project/list in my task app, will my next push respect that or overwrite it?

For the destinations the app manages, the app's settings win — if you rename the managed project/section/list inside Todoist or Google Tasks, the next push renames it back to match your app settings. The app reuses the same project/list by its internal id, so renaming is done in place (your existing tasks aren't moved or duplicated). If you want a different name, change it in the app's card, not in the task app.