I was sort of wondering where the Norbert Weiner reference in your permalink came from…
What you’re referring to are Twitter Cards which use custom (invisible) metadata from your website’s header section. Your site that doesn’t display them just isn’t giving any data to Twitter to parse to include it in the tweet. For most WordPress sites the metadata is typically coming from one of many plugins. Most commonly it’s either All In One SEO or Yoast though there are JM Twitter Cards and a slew of others. (Ideally you’d use just one at a time to cut down on potential duplication or even conflicts.) I’ve been using All in One SEO to handle this (and other things) and it typically uses some smart defaults (post title and either the excerpt of the first few lines of the post’s body as well as the featured image) if I don’t go in and change them manually. Within the “social” tab of the plugin’s meta box it allows me to manually change a few parts of the Twitter card including the title, the synopsis, and whether the card uses a small image or a larger one. Some plugins also let you do video and other Twitter card types.
To get things working you should check the details for your plugin of choice. It typically requires you doing a simple white-listing of your site with Twitter to register that your site knows how to handle specific card types before they work properly. If you need it there’s lots of documentation on Twitter as well as a validator tool to test things out.
Hopefully this, a bit of reading, and some experimentation will get you the results you’re looking for. Otherwise feel free to ask.
I don’t remember where I documented it, but I’ve been known to use the extra data in Twitter cards to visually extend the 140/280 character limits Twitter imposes. For bookmarks, reads, and other short link type posts, I’ll often use the “body” of the Twitter card to add my own two cents to what I read in the syndicated copy on Twitter rather than adding it in the main body of the Tweet. Sometimes I’ll use it to add additional context. Often I’ll also use an external image (or the one parsed and included via the Post Kinds plugin) to decorate a Twitter card while my site doesn’t explicitly show an image in my version of the post. For some research related sites I often bookmark, I’ll save a small logo sized image in my images folder for use so that I can quickly use it and not need to bother with custom images on each post [example]. Depending on the plugin, you can typically upload a default photo to use if you don’t explicitly set one. In my case, it’s that image of Norbert Weiner at the chalkboard, though occasionally I’ll change it for fun. I suspect you might use some clever Lego image?
As a fun example of how various silos create and use their own metadata standards, Kevin Marks wrote a post about Decaying Silos as dead malls for which he carefully marked up the page so that when you share the post on various platforms, you get customized, appropriate, and funny previews of each of them specific to each silo.