Is it possible to watch the summary story that is told during the install of Starcraft 2?
Solution 1:
The video is available in two parts on this Join the Dominion! website. The two parts are called "New Era" and "Uncertain Future." As of this writing, they were easily accessible on the right side of the linked site.
Solution 2:
For what it's worth, the starcraft 2 manual comes with a 5-page, "extended" version of the in-game history, covering things the installer leaves out, such as the UED invasion. (i.e, Brood War's storyline). The text can be found on pages 8 - 12.