Tanha is about wholesome and unwholesome desires, ie those that lead to or dont lead to liberation. Its not about desires that do or do not change you, as this article is categorizing it.
Screenshots preserve exactly what the user sees, and for this reason i believe they are the best format for evidence. They also preserve their look when your messaging software starts doing funny things with it. They are immune to accidental edits. They are reliable in the sense that if the original source ever changed, no one would argue that you didnt see what you saw with a screenshot. They wont degrade when the content of a url changes.
We do use screenshots extensively, but alongside links and copied text when necessary