Description: Defines a description of the document
The "document" in this case is a specific tweet which can best be described with its content since it is a self-imposed summary as-is. In short, you're wrong.
This is one of the things that I feel is most misunderstood in HTML, the page's information should be specific to that page where possible.
It's tempting to have a description that describes all your pages simply for easiness (hell W3S goes ahead and does it wrong in that example) but it should define what the page is about, an abstract of a document if you will.
w3schools is not an authority and has no affiliation with the w3c.
With that said though, I see your point. I still stand by the fact that Google should be the one to parse this information and Twitter shouldn't be preparing it specifically for Google. Similar to how some YahooSearchMonkey developers are doing it with yahoo:
The META element can be used to identify properties of a document (e.g., author, expiration date, a list of key words, etc.) and assign values to those properties.
No matter where you choose your source, meta elements apply to the document. The document is not the web site. It is an individual page such as a tweet, a user's profile, or the homepage.
Why should google go out of their way to fix shortcommings of anothers website, twitter will be brought extra traffic by those not using twittersearch and google has nothing to gain by replacing twittersearch.
Twitter's goal should be "to properly use html tags/properties in a way that does not ruin search results". If Twitter provided no meta description, Google would be forced to create their own description, which you could then blame them for getting wrong. Instead, Twitter uses the tag inappropriately, which Google trusts.
This isn't Google's problem; it is Twitter's. Whether or not Twitter cares is a different story entirely.
Whether twitter cares or not, Google should care. That's what this issue boils down to. It's a problem with the Google user's experience, not the Twitter user's experience, despite who's fault it is.
Description: Defines a description of the document
The "document" in this case is a specific tweet which can best be described with its content since it is a self-imposed summary as-is. In short, you're wrong.