Very briefly, the idea of a natural kind term is that of a term that “carves nature at its joints” (Plato’s metaphor), ie is not just a convenient way of splitting the world up into entities but one that reflects the true structure of reality.

Philosophers have debated various criteria for determining what constitutes a natural kind, which I think would have added some clarity to the article.

A related approach that could have added clarity (though I can’t recall whether it deals with natural kinds per se) is Hacking’s Social Construction of What? which lays out a sort of hierarchy for how socially constructed a given phenomenon is. It attempts to add rigor to our intuitions around how objective or subjective a given judgement is.