I installed an Ubuntu alpha or beta and have been keeping it up to date.
So, Should I presume that my system has automatically updated to the next alpha or beta, and will be running the official release when it's finished?
The beta uses the repositories as they are updated -- it is not a fixed snapshot of Ubuntu as it was when the beta was released.
If there are no more updates to be installed, then congratulations! You are running the final Ubuntu release. There is nothing else you need to do.
Here, have a totally simplified, consumer-side view of a release I hacked together in ten minutes:
In this chart, when you run