I can define a female and woman quite well.

Sex is a continuous bimodal distribution. Every sexual dimorphic trait follows a distribution. For example, the penis and the clitoris are the same organ, and organs can grow in all forms in between. Still, the vast majority of the population is born with something on one end or the other of that distribution. Same for all sexually dimorphic traits. If most of your sexual traits are towards one end of the spectrum, your sex is male, the other female.

And gender is on top of that. Society applies various values to the two sexes. Society classifies certain things as masculine, certain things as feminine. Society encodes many non-physical traits, behaviors, cultural practices, activities, etc. as masculine or feminine. A man is someone whose various gendered traits primarily align with the male sex cluster. A woman is the opposite. If you mostly want to do the male-coded things and be seen and treated as male, then you're a man. If you want to do the female-coded things and be seen and treated as female, then you're a woman. Having a few masculine hobbies doesn't make someone a man. We're talking broad averages here.

Or more succinctly.

Sex is a bimodal distribution. Culture creates two big blobs of traits mapped to each end of the sex spectrum. A man is someone who exists or wishes to exist in one of those cultural blobs, a woman the other.