Well, shucks... You've already found The Trademark Guidance Page. So I'm really not sure where the confusion comes in. I mean, there are some fairly detailed do / do not guidelines there, but even if you stopped reading after the third section you'd have seen:
So if you were making a product, and you used a Stack Exchange name or logo in your product (or in its advertising) in such a way that would mislead someone into thinking that your product was owned by, operated by, endorsed by, or in any way part of Stack Exchange Inc., you would be violating the trademark and this would not be legal.
...which is pretty clear on disallowing the use in situations where it might lead someone to think we had some connection to your product. After which you'd have read:
Our logo images and site names are copyrighted. Any content on the Stack Exchange Network not contributed by users is copyrighted. Copyright is different than trademark. Ordinarily you couldn't copy it. But if you were writing a news story or blog post about a Stack Exchange site, reproducing the logo image would almost certainly be considered "fair use" and allowed under trademark law because you were not using it "in trade."
...which I would think pretty much settles it: you're not writing about SE. You're not parodying SE, or really even talking or alluding to SE in any way - from the sound of things, if I'd never seen or heard of Stack Exchange and saw the logo in your game, I'd just assume you came up with it on your own...
Of course, if you read further, you'd have found this:
We generally do not permit use of our name or logo on merchandise.
- Do contact Stack Exchange Inc. with a detailed proposal if you would like to request an exception.
- Do not use the names or logos owned by Stack Exchange Inc. on any apparel or merchandise without our permission.
Normally, this refers to slapping the logo on tangible goods, but as a "collectible item" your in-game use is arguably a simulation of a physical piece of merch.
So... Don't use it. If you must, then ask first.