It appears that the method for renaming the Heroku database has changed since the last update; The answer specified in > this answer is no longer valid.
I spoke with Heroku support, and in the current way to rename the Heroku database, the heroku addons:rename command is used (you can find a quick CLI help on it with the heroku help addons:rename command.)
For example: if your database is called postgresql-loudly-9983 and attached to the bumping-softly-6892 application, and you want to rename it to stats-db , your command will be as follows:
heroku addons:rename postgresql-loudly-9983 stats-db -a bumping-softly-6892
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