There is no email that is sent to a user when their profile is updated. That's only available through "CB Profile Update Logger", but it by default only notifys CB Moderators. If it's notifying the user being edited they're likely a moderator. Review what you've Moderator View Access Level set to within CB > Configuration > Moderation to ensure you've set it to a strict view access level and isn't set to something like Registered. Another option is if you configured an auto action in CB Auto Actions to deliberately do this.

That's probably the users Welcome email, which can be configured within CB > Configuration > Registration. Changing it or removing it there will affect all users and not just those you create in backend.