Funnily enough I did two years of postdoctoral research on categorisation, albeit from a cognitive science and not a philosophical perspective. That was 20 years ago and I didn't continue with that area so my memory is a bit hazy, but I know that monothetic intensional definitions really didn't feature. We have something called the defining attribute approach which corresponds to a monothetic intensional definition. The first thing taught about that is how useless it is for explaining how people use categories in real contexts, and the example given is the one Steersman keeps using: bachelor = unmarried adult male. According to that definition, an 18-year old unmarried male student and a priest who has taken a vow of celibacy are bachelors, but nobody would use that term for them. Intensional definitions can be useful for something like a machine learning algorithm, but also resonsible for the failure of output to resemble human cognition in AI applications. Granted, that is referring to 'everyday' categorisation and not scientific classification. But monothetic definitions are not always possible, and polythetic definitions can be formalised to make intensional definitions and are used that way in many areas (like some diagnostic categories, and in fungal taxonomy for example). In previous posts he seemed to equate polythetic definitions to 'family resemblances'. Family resemblance refers to what in cognitive science is called the prototype approach, which is usually considered the best description of how people do 'everyday' categorisation. The prototype approach is polythetic in that multiple attributes can be considered intuitively in comparing an exemplar to the prototype, but a formal polythetic approach can specify specific numbers or combinations of attributes from a larger set as necessary and sufficient for category membership.
I'm taking a guess as this is way out of my area, but possibly polythetic definitions are used for fungi because they change so much during the life cycle that it might not be possible to have one set of attributes always present for classification? Having several alternative sets of attributes that can be used for classification is quite valid as long as they can be shown to result in correct classification. This could be demonstrated, for example, if there is one feature that is always sufficient for classification when present, the feature is not present at all times during the life cycle, but that other specific combinations of characteristics can be shown to accurately predict this feature.
Excellent post. "The clarity is devastating!" Thank you.