nouns - Terms for "natural gender" and "grammatical gender"

This post is partly inspired by previous posts, such as
this one, on non-existence of grammatical gender in English. My question is mainly about what "natural gender" and "grammatical gender" are to an English noun.

There are nouns, such as 'mare', or (debatably) 'ship' whose natural gender is perceived to be feminine by significant number of speakers. Most nouns, however, either are of a neutral natural gender or have got no natural gender. (I wonder whether there is a consensus on which of the two is the case.) Wikipedia also seems to suggest that gender pertains to referents rather than to nouns.

In the linked question, the OP derives "grammatical gender" from
"natural gender" for those nouns that have got the latter. This is relevant, because a natural gender seems to be the sole reason to even think about a grammatical gender, or traces thereof, for a word. But then it seems to entail something like a partial grammatical category that only some nouns have. It is not inflectional. It is not a remnant of previously existing grammatical gender in English.

Is there any term for what "natural gender" and the putative "grammatical gender" is to a noun?