We'll be withdrawing the support for webhooks before the end of the year, to improve reliability. Removing the webhook support will also avoid tying our service to specific git-hosting services.
You can still use GitHub, BitBucket, or any other site to host your primary git-repository, it is only the update-mechanism which is no longer tightly coupled to those particular services.
To convert you need to do two things:
Removing the webhook configuration should be as simple as going into your GitHub/BitBucket repository settings, locating the webhook-section, and removing it.
Once the webhook has been removed you may then access your local repository, and add the remote - as per the settings on your profile page. This should be as simple as:
git remote add dns-api firstname.lastname@example.org:$USERNAME git push dns-api master
Once you've done that any future pushes you make the the dns-api-remote will result in your DNS changes taking effect.
In the past your pushes would become live when the webhook ran on your behalf, in the future your changes will become live whenever you push explicitly to us.