I don't think "strong", "scared", and "curious" have fixed definitions at all. They are subjective, and one person doesn't always understand what another means.
For example—why did you put "strong" alongside "scared" and "curious"? From that context, one could have assumed you meant emotional strength. Different psychologists have different things to say about emotional strength. But in your follow-up comment, you refer to physical strength. A physical therapist has different hopes and expectations for a baby, a teenager, and an elderly person, depending not only on their age but on whatever individual injuries or limitations they may have that put them in physical therapy in the first place. A zookeeper will warn you that an elephant is strong because it can step on you. An entomologist will tell you that an ant is one of the strongest animals on the planet because it can lift 50x its own weight (a grain of sand). On average, men can lift a little more weight than women of similar sizes/ages, but it's only an average and there are many individual exceptions to this, especially when matching up a woman who exercises with a man who doesn't.
Similarly, I can be "scared" or "curious" in different ways. I can be "scared" of getting married, of dying alone, or of pit bulls. I can be "curious" about the works of Shakespeare, what would happen if I asked someone on a date, or whether it will rain.
These words and possibly even these concepts will not have exact translations into other languages. There may be cultural differences.
They are definitely context-dependent. If you ask me whether I'm "scared," this is a reasonable and comprehensible question to me if there's a zombie horde knocking at the door, but I might look at you funny if it's a sunny day and your question comes out of nowhere and seems to refer to a box of chocolates. (You might be having an anxiety attack that feels obvious to you but isn't legible to me.)
People do often care if you make comments about whether they have these sorts of emotions or characteristics. If a student says they're struggling with terrible "self-doubt" over their ability to write a dissertation and finish their program, and you say, "Hmm, I don't know what 'self-doubt' is, can that be broken down into its component parts? Maybe you're scared of the work, not strong enough, and lack curiosity," they will be upset—not only because you've insulted them, but they may disagree with you because your criticism may not be well-informed. They may have a different self-understanding which comes from a wealth of facts about their own situation. (To them, "self-doubt" is contextualized by their financial problems and family distractions, without which context no one else can begin to understand their other feelings about themselves and their academic work.) Now their concern is that you may actively try to impede their progress (e.g., by spreading rumors and excluding them from group work) because you're questioning the meaning of the feelings they do/don't have and you don't take them seriously.
Regarding gender—to the extent that it's a construct, it could be otherwise, so it isn't "necessary" for it to exist, but meanwhile it does exist. And "it" isn't just one thing. It means different things to different people.
There isn't a single definition of what it means to be a man, a woman, or something else, nor is there a firmly defined scope of "gender," and there doesn't have to be one.
Regarding your example in your article of women reporting feeling vulnerable walking at night: This kind of response is context-dependent and will depend who you ask. 20-year-old women on an urban college campus who do their hair and makeup and walk to frat parties where there's lots of beer have a certain set of experiences and assumptions about walking at night. A 25-year-old woman who's grown up in a patriarchal Muslim household in Afghanistan has perhaps never left the house alone at night; her father and her society may not allow it, and there's nowhere for her to go at night anyway. Similarly, a 15-year-old girl who got pregnant and was locked away in a Catholic convent has not had a chance to leave the walls in years. This is different from the experience of a 30-year-old, six-foot-tall, short-haired cowgirl in the desert who rides a horse and carries a pistol and is often mistaken for a man when she goes to the saloon. Her questions about vulnerability are is there moonlight so my horse can see where it's walking and do I have bullets.
What it means for any given person to "feel like" a woman or a man is probably not a single objective trait, but a collection of traits and, more importantly, their subjective interpretation of those traits—not so much whether they are "masculine" or "feminine" traits for anyone/everyone who happens to have them, but what it means to them personally to have them. Maybe feeling "masculine," "feminine," etc. in a particular moment is similar to an emotion like "scared," "curious," etc. — the self-report doesn't have to be demonstrated or proven but could simply be believed. Whether we believe each other may be the operative idea here. Maybe when I feel "scared," I get goosebumps, and when you get "scared," you sweat, so "scared" does not physiologically feel the same to each of us, and yet we could just believe each other if we say we're scared, because that's a starting point for communication. Similarly, "masculine" may mean different things to us in different contexts (or nothing much at all), but we could provisionally believe each other when we say we do (or don't) feel masculine in a particular moment, because thus we begin to listen to each other. Feeling "like a man/woman" in life overall is a broader, more enduring form of self-interpretation, so, if we question someone's self-assessment of that (which would be unfortunate), we're showing skepticism not of a single feeling/experience but of thousands of collected feelings/experiences, which is essentially what it means to them to know themselves as a person, which is why it offends them to have that questioned.
The problem you've set yourself up for in your article is at the beginning: “The current precedent is that self-identifying as a particular gender is absolute, and we should accept the individual’s claim without question.”
First of all—transgender people often say complex, nuanced things about gender. Not everyone says self-identity is absolute, whatever that means (the absoluteness of "man"/"woman," and/or the absoluteness of one's ability to self-know it?). I don't think you'll find agreement on whether "self-identifying as a particular gender is absolute" is in fact established precedent, because the statement is so vague I don't think you'll even find agreement on what it could mean.
What I see as a bigger problem here is the implication of the stakes: accepting or not accepting someone's statement about their own identity. If you set yourself up as the arbiter and essentially say, Why should I believe anything you say about yourself? Why shouldn't I reject your self-classification and classify you however I like? Why don't I have social power over you? you are creating a context for the sort of answer you get. Especially in the broader political context of extreme anti-transgender laws, if you question someone's ability to know their own gender, you'll get a defensive political answer, not a real philosophical exploration of whether gender is absolute/relative, objective/subjective, universal/cultural, personal/social, how gender is or isn't affected by physical embodiment, etc. Trans people are living under innumerable, ever-changing political threats right now, and that affects the way we do philosophy with cis people who are "asking questions" about our realness. If you question someone's self-knowledge in this way, you are (intentionally or not) appearing to align yourself with the forces that threaten their legal autonomy, so of course they're going to give a reductive answer like because my gender is what it is, and I just know it. You are the one who demanded a clear, simple definition, so now they are giving you the oversimplified definition you said you wanted/needed as a prerequisite for accepting them, in the hopes that you will accept their gender and leave them alone. If, on the other hand, you start with the assumption that transgender people are capable of nuance and self-knowledge and that we should have legal autonomy regardless of whether our self-understandings and our vocabularies are simple or complex, and if you don't place our social recognition and inclusion at stake at the very outset of a philosophical discussion, then you will hear all sorts of interesting ideas about what gender is and isn't. We can be anti-ontologists who don't reify our genders and yet we can also go through gender transitions and have hormones and surgeries related to that and we can be recognized as members of our gender and have human rights. We do not have to articulate excruciatingly simple dictionary definitions of what gender is to be entitled to live transgender lives. Nor do we have to agree with every other transgender person to be allowed to live our own transgender life.
You should, in fact, accept everyone's claims about their gender/sex (whether they are trans, cis, nonbinary, intersex, or don't want to tell you exactly how they are or aren't man/woman, male/female), and your acceptance shouldn't depend on their offering absolutist oversimplified definitions of what these words mean. Allow space for people to use common words to express nuanced things, and you'll hear them say nuanced things.
What doesn't work is the self-contradictory demand for ontology and anti-ontology, reification and non-reification, subtlety and straightforwardness, nuance and clarity. Especially as a prerequisite for just being allowed to say "Hello, I'm a man" or "Actually, I'm a woman." A trans person can't give you all that self-contradictory philosophical background. You can just start by accepting their statement that they are what they say they are. They are trying to live in a world that is trying to make them illegal. You can do the self-exploration of your own gender (or lack of gender) without positioning yourself to undermine someone else's self-exploration. If you don't understand what a particular trans person is saying about their gender, that's OK. They're still the expert on their own gender. You might understand a different trans person. Whether you personally relate to what a trans person says about their gender might have nothing to do with whether you want to be friends or colleagues with them. Sometimes you might not understand what a particular cis (i.e., non-trans) person is saying about their gender, which is also OK. The cis person is also allowed to be the expert on their own gender. People who don't care very much about their own gender are allowed to say they don't care, and they can like or dislike the term nonbinary for themselves, and they can hate gendered pronouns or not care about gendered pronouns, and maybe they're fortunate to speak a language that doesn't have gendered pronouns, which is probably a better linguistic construct that avoids a lot of problems and useless arguments that are regrettable byproducts of English.
