How can I create an “excerpt” with text that won’t be displayed in the post itself?

It’s not. I believe when you go to make a post, there’s a box just below it that says “Excerpt” Whatever you put in that box will only show up as the excerpt.

I tested it out and it works. You can view it on the website i’m currently developing here: http://www.kirschfamilycomputerservices.com/sam/wheresthefairness/

The first post. Note how the excerpt is different from the post itself!

Also, if you want to run it in a loop instead of making a post I think there’s a way to do that, too. check here: http://codex.wordpress.org/Excerpt

Good luck and hope this helped.