The button appears on the Birth Record after saving it.
After clicking the button, you will be redirected to the newly created dog record. The dog record will have a name generated based on the Birth Records record id. Example "Puppy - 2". When the dog has been named, you can edit it's name from the dog record and it will also update the name on the birth record.
If you entered Gender, Birthdate or Breed incorrectly on the Birth Records record, and have already created or related it to a Dog record, you will have to change the information on the Dog's record before being able to correct it on the Birth Records record.