My five year old knows the difference between a man and a woman, I'm not sure why people find it so hard to work out.

And at the point one of the 0.5% of the population that's transgender suddenly pops up i'll tell my five you old about gender dysphoria in a way they can understand.

If they were "just words", they wouldn't be informing public policy so much as present.

It matters because society does not shape its public policies based on both of those definitions (sex and gender) combined, but based on one or the other definition strictly (sex or gender). When the subjective definition (gender) becomes the prevailing one, public policy problems are had, I feel, in the sense that, for instance, single-sex spaces now get disingenuously redefined as a form of discrimination against "certain women" or "certain men", as applicable, with the negative consequences falling one-sidedly on women (by which I mean biological females) across the board.

This for me isn't about the semantics of which pronouns to use. It's about whether women specifically will have such basic entitlements as the right to privacy, to safety, to fair play, or to freedom of speech and association going forward.

Yer when I read it there was a lot of crossover. I think you need a 5th Agender. Seen some articles (mine you i didn't research to see if they were true or not), where male and feels cut of their breasts (males nipples) and mutilated their genitals. I guess they fit under that category.  Even if false, I guess it could happen, where someone decides to do that. 

"These four should be sufficient. All the others are people who struggle to find  their identity, so they try to push themselves in categories. I don't think this is very useful (especially since a lot of them basically mean the same thing), and I also don't think it's a very healthy thing for these people to do."

so if someone wants to be a unicorn, for example, the other people around them should be expected to play along with that?

A collection of stereotypes of male and female behaviour. Which happens to be for 99% accurate. Since less than 1% of US is population does not identify with the gender they are born with.

Some of the older trans-identified people I know reject the concept of transgenderism though. They believe that since they've undergone surgery to have their birth genitilia either radically altered or eliminated, they now, in fact, belong to the opposite sex and that trans-identified people who haven't undergone such medical procedures are mostly, if not entirely, opportunistic frauds. These people refer to themselves as transsexual specifically and reject the term transgender.

These are the sorts or reasons why the lists of "gender identities" are as lengthy as they are.

I don't know about that. There's a heavy age skew to that data, with younger people being significantly more likely to identify themselves as trans in some way. There's also an interesting skew in terms of sex. Apparently 73% of trans-identified people overall are biologically male, while about 70% of trans-identified youth are biologically female, and more specifically would otherwise identify themselves as bisexual or lesbian in terms of orientation. These types of statistical skews hint at the superficiality of gender identity as a concept in my mind, and at minimum merit further scientific investigation, I believe.

Bottom line: I think the popularization of transgender politics of late is mainly owed to social media, that it is a growing trend that will increasingly affect public policy going forward, and that more and more people are going to identify themselves in these sorts of ways as this trend continues. Don't bet on that statistic remaining the same for long.

Depends on how serious they are about it.

But what does that have to do with what I said?

I think the best distinction alongside which we can separate sex and gender is the material and ideal distinction.
Sex is our own material reality. You are born physically a man or a woman (or intersex in exceptional cases).
Gender is the ideal reflection of that material reality. What social structures are build around us being male or female? What is expected of us as a male or female? How are our individual thoughts structured in relation to the gender structures of society.
The problem we encounter is the same as the use of the words "race" and "ethnicity". They're being used interchangeably, leading to people getting confused and even angry over the concepts.
People have especially poor ideas around what gender as a constructed reality means. People thing "Well, if Gender is a construct, I can be/ choose to be/ make the term out to be anything". leading to confusing unscientific ideas like the gender spectrum, which disassociates the gender idea from the material sex, rendering the term practically useless.