I consider “fake news” to be erroneous or shaky stories that are being passed off as truthful, verified, and proven, news stories made up to garner ad clicks by people who are trying to gam Google, or stories distributed by blogs/partisan sites/etc. that lack any standard of objectivity or journalistic ethics. Those include completely bizarre, out of the blue claims, as well as stuff that *almost* sounds true.
Colbert was pure satire with a huge dose of absurdity — and while some truths may have snuck in there (depending on your perspective), I didn’t sense he wanted to be treated like a news outlet. The over the top persona made it theatrical.
Stewart made plenty of jokes, but his was more of a comedic commentary on events than a satire. There were satirical segments (usually the remote ones), for sure, but he was being himself, and sharing what he actually thought (and he dinged both sides of the aisle, even with a clearly liberal perspective).
The reason people call those shows fake news is because that’s their shorthand for satire, or commentary on events they disagree with. I think Colbert was satire.