Just to clarify, running this command:
npm install -g ionic@beta
is updating the Ionic CLI Version not the framework. So when you run ionic info in your project folder and get something similar to this:
Ionic Version: 2.0.0-beta.3
Ionic CLI Version: 2.0.0-beta.21
then run the above command, it will only update the Ionic CLI Version which is used to run the ionic commands i.e. ionic start, ionic serve, etc.
If you want to update your version of the framework, you can modify the package.json where it says ionic-angular to the version you want, and then run npm install in the project directory:
"ionic-angular": "2.0.0-beta.3",
More info on this here: http://ionicframework.com/docs/v2/resources/using-npm/