When a browser is pre-caching, it's loading the links on a page before you actually click on them -- the idea is that if/when you do click on one, it's already got it loaded and can display it right away. If the link "does" something, though, like delete a picture or album, then is the browser fetches the link, it's just like you've clicked on it. By logging out, the links aren't on the pages, so they can't be loaded.
I have no idea why you'd see it on different computers and browsers, though. That's really odd, since it's unlikely they would all be set up with pre-caching. That has me stumped. Were all those computers going through the same internet connection? Maybe you have a pre-caching proxy server on your connection.