It cannot be done using this approach, at least not with true certainity. All you can do is track if 1) the window was successfully open using onclick and 2) if the window was closed (you can assume that perhaps the user completed the tweet).
You will NOT able to get ANY information from you open window via Javascript because the window is NOT in your domain (meaning, it is not on your server but on Twitter's).
What you want to do is implement this functionality in code as opposed to a javascript popup window. I have seen a number of commerical products that do exactly what you are describing.
The documentation (links) you provided refer to the REST API for Twitter (which usually works on the server) that is VERY different from Twitter's Web Intents which is client site javascript plugin, which has no similar API.