Is this an error in the documentation?
"To edit a page you must have at least the permissions obtained by being added to the default<Site Name> Members SharePoint group for the site."
The "Members" group is granted "Contribute" role permissions. But the "Add and Customize Pages" site permission is required for adding or editing a CEWP, and is not included with "Contribute". It is granted with "Design" or "Manage Hierarchy" roles.