Now, having addressed the opening of your reply, I'll get on to the more respectful remainder of it, in a similarly respectful tone. I appreciate you reading it (and apologise that I've not had time to reread it myself before posting the below, & am rushing a bit, so hope I'll find my arguments stand on revisiting tomorrow - not ideal, but a fast-moving thread & little time!)
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Of course the way that sexuality manifests is culturally dependent - we didn't even have language for it until recently. I don't think anyone would argue otherwise, but even this writer appears to acknowledge that gender identity is innate, regardless of how it manifests. S/he then goes on to conflate gender non-conformity with being transgender. This is the biggest failing in this article. Gender expression does not always map one-to-one with gender identity. You can be a gender non-conforming women whether you are transgender or not.
I think you somewhat misunderstand the premise and detail of the article, and that this is reflected in your own unconsciously ironic adoption of a range of terms the author argues are wholly socially-constructed & subjective to support a (mis-)interpretation that relies on these terms having commonly understood, concrete consensus meanings.
As such, I'd say that a more effective rebuttal from your perspective, acknowledging your values, may be, for example, a scientific, peer-reviewed source that offers a widely accepted definition of "gender identity" and "transgender" (as opposed to, for example, "gender dysphoria"). I'm genuinely interested if there's one out that that covers adolescent girls, cross-dressing men & the clincally dysphoric.
Even if you do, the author would point out - does infer - that science evolves over time, citing the medical consensus on how PTSD, for example, typically presents, and associated shifts in diagnostic criteria. It's interesting you read this as an omission: that s/he "ignores the fact science getting better and better at understanding mental health". I see this as implicit, but secondary to the wider argument that science & sociological phenomena intersect in complex ways.
His examples of past interpretations are evidence for, not evidence against gender identity. He doesn't call it a belief, at least, it's just in the normal range of experience.
I find this a little confused. You appear to be suggesting that the exact 20th/1st century phrase and concept of "gender identity" is supported by examples that, the author him/herself is arguing, instead suggest that "our" two-word phrase is, rather, a reductive contemporary categorisation of something much more complex & fluid, that manifests in a wide variety of ways across time and space. I find this an open-minded acknowledgement of the breadth of human experience.
He then argues that transgender people don't exist - which absolutely does not follow from what he said before.
In the light of what I say above, this is a surprising reading as you, again, assume and impose current terminology, with all its potential limitations, on human history and world geography. I suspect that the author may argue that there's a certain arrogance to metaphorically sticking this somewhat simplistic label on to all peoples, in all places. I'd argue myself that it dissolves important distinctions into a limited westernised conceptualisation of something far more complex. From this perspective, I may even tentatively suggest that your own alignment of the fa’afafine with "gender identity" could, itself, be seen as something of a culturally colonialist imposition.
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Certainly, it's very interesting.