It's entirely context dependent. For me and the people I deal with (which deal with various wikis in our lives for both work and play), "the wiki" refers to the specialized wiki on the topic we are speaking (for work it's the work wiki, for the game were playing it's the wiki for the game, etc), and Wikipedia is used to refer to Wikipedia, because it's not specific to anything. If someone just mentioned "the wiki" or "wiki says" in a conversation and was intending to imply Wikipedia, it would just confuse me by making me think there's some specific wiki they were referring to that I missed previously or they weren't clear in communicating, unless we were specifically referring to Wikipedia earlier.

The humor tag is there because the community is self-aware that it's mildly pedantic. The views of the essay are held by a strong majority of the community at least. Nobody ever abbreviates Wikipedia as "wiki"; we have far too many wikis floating around and it would be confusing.