That article seems to say that the standard definition of "monkey" does not include apes, and thus humans.

It doesn't just seem to say it, it says it explicitly: "monkeys are, in terms of currently recognized taxa, non-hominoid simians". Perhaps the accepted terminology may change at some point, but currently apes are not monkeys.

I remember reading or hearing that if we follow taxonomnic rules from the ground up, humans would be classified as hagfish (don't quote me on that, I have a terrible memory)

We've not made much progress on this front since Plato's featherless biped.