WordPress.org

I don't see a new orange button below the normal "Download" button on WordPress.org like in your screenshot. What gives?

Did you make sure to install the Greasemonkey script? It's what adds the button to WordPress.org. To install it, go to your admin area and visit Plugins -> WP.org Installer and then click the button there.

Can I use this plugin with multiple blogs? Will multiple buttons show up?

Yes, you can and they will!