Gender is very difficult to explain because a lot of the visible examples of it can actually be explained by gender roles.
I do agree that there are some biological differences due to different hormone levels, but these differences will also vary from person to person, and aren't a good way of describing what gender means, I don't think.
I'll be honest--it's hard for me to understand gender. I don't know exactly how I feel about it. I identify as genderqueer, and I've struggled with dysphoria to some extent for years, though to a milder extent than many FTM people do. A lot of the dysphoria is physical, and I think I would be happy to have a male body. But that's not the full extent of it, either.
When I try to explain identifying as genderqueer or trans, it seems like a lot of people are quick to assume that I'm a woman who can't cope with having masculine interests or traits. Yet in reality, I don't see myself as that stereotypically masculine at all. I played with dolls as a kid, and my writing is very focused on relationships. If I were a cisgender man, I don't think these things would make me doubt my gender identity. As it is, there's pressure to be stereotypically masculine in order to "really" be trans.
I don't think my actions or habits define my gender. I stopped shaving my legs and underarms a couple years before I started to seriously question my gender identity, and it seems strange to me that a grooming habit that was once "transgressive" is now something that's expected of me if I'm ever to be seen as male. Yet, it's challenging to talk about my own identity without using gender roles as examples. I feel more comfortable in male clothes, but there's no reason why a cis woman can't feel more comfortable in male clothes. And at the same time, some gender roles (male or female) simply don't appeal to me at all. I never want to become a guy who objectifies women, for example, or feels pressured to do macho things. Being male in our society will never be a perfect fit for me, but it's also not a perfect fit for a lot of cis men.
When it comes down to it, I don't know what makes me male, female, or none of the above.