You CAN objectively define a word, IMO, as it would be dependent of mind, but also dependent on subject. By 'objective' all (or a majority) minds would conceive the same (or similar) subject.
The main caveat is frame of reference.
The point I thought you might be getting at is technically objectively defining a word as a means of reaching an objective, in this case it could perfectly apply to the gender scenarios because you can set an objective of defining a word for the purpose of categorizing on appearance.
So man could be defined as "male appearing" with the objective of differentiating between woman being defined as "female appearing". But this is why I draw the distinction between sex and gender because one can appear as a woman, and statistically be viewed as such on a first assumption, but lack the mind-independent chromosomes that dictate the differential between a male and female.
Now I see where your comment is getting at, I personally don't agree with that metric for objectivity, I know there are people that hold to that, but from what I have observed, I wish to avoid the possible logical pitfalls of that position.
