I very much have personal experience with 'woman' as it applies to more than half of the population. Not one person, government or organisation has ever needed to know what my chromosomes are, my internal organs look like, what gametes my body was organised to support, or the status of any SRY genes. I live under the same language, law and cultural beliefs. I am not above it, nor do I control it.
I do feel that things like this highlight how limited, privileged (and, ironically, sex-based) your conviction that there's no need for a shared noun for adult human females is.
Maybe think of it this way...
Your definition of "woman" liberates you to say, as you do here, that "I very much have personal experience with 'woman' as it applies to more than half of the population."
But you saying that renders the same phrase meaningless for nigh on all of the 4 billion women globally. Because, in their wonderful infinity of collective experience, they'd be mystified by what it could possibly mean.
Prior to your inclusion, of course, they'd have been perfectly clear: menstruation, a likely capacity for pregnancy for a proportion of their life, and menopause. With your inclusion, though? That's the silence of not far off 4 million women scratching their heads in bemusement as they try to work out what commonality could possibly unite them all outside of that. I mean, we're so wonderfully, fabulously, expansively different! What's left, besides biology, that doesn't flatten "woman" into unrepresentative homogeneity?
Your answer would be, I assume, "being seen and treated as women". And right there is the privilege. Because that's an extraordinarily dangerous definition that women simply can't afford. To be defined by how we're seen and treated? Really?! Look around you at the world today!
Your definition downplays the physical reality that is the axis of our shared oppression to indiscriminately embrace the very constructs which exploit that reality.
Your phrase "Not one person, government or organisation has ever needed to know what my chromosomes are, my internal organs look like, what gametes my body was organised to support, or the status of any SRY genes." is so extraordinarily and ironically revealing. We'd kill for this to be about some theoretical "knowledge" of our gametes and chromosomes etc. for us. If bloody only! For women, knowledge is irrelevant. Our life is the experience of living them. To misquote Descartes, We "know" because we are.
Your more liberal and liberating definition? Our prison.
Your conviction that women globally can afford (let alone want) to embrace woman-as-perception over retaining a shared word with which to name and resist the daily - hourly - challenges that unite us? Our evidence of your privilege.