How to determine which commit and/or which branch was deployed to Heroku?

To see what's been deployed on the Heroku dashboard: Just click the Overview tab. You'll see an Activity view on the right that shows recent deployments with commit hashes.

The "releases" feature does not help you because it's an internal counter Heroku is using to keep track of pushes and re-configuration.

If you re-configure your app, the counter will increase. Deployments will also increase the counter but that doesn't help you.

But remember that Heroku is just another "git remote." So if you are on the machine which you used to deploy, there must be this:

$ git remote show heroku
* remote heroku
  Fetch URL: https://git.heroku.com/your-app-3367.git
  Push  URL: https://git.heroku.com/your-app-3367.git
  HEAD branch: master
  Remote branch:
    master tracked
  Local ref configured for 'git push':
    master pushes to master (up to date)

If you have a similar output, you can just:

git show heroku/master

and see the commit that was pushed and deployed most recently.