Okay so, as a queer biochemist with a penchant for etymology (study of words/their origins), I feel almost uniquely suited to explain the "problem" with arguments relating to the cis/trans prefixes, but the first thing i want to say;
when discussing cis men and trans men, you're not actually using the words as "cis" or "trans." again, they're prefixes. the terms are "cisgender" and "transgender," as in, you have remained the gender you were at birth or you have another gender to the one you were assigned at birth. it's like how "homo" and "hetero" are, admittedly occassionally, used as shorthand for homosexual and heterosexual. but yeah, it's why "trans woman" is correct, and not "transwoman." because trans woman is actually shorthand for transgender woman, like how you could shorten "expectant mother" to "expectant mum."
now, for the term, "queer." while it originated as a slur against the lgbtq+ community, some chose to reclaim it, and it has now taken on status as a sort of umbrella term, similar to gay can be used as an umbrella term for lgbtq+ people. of course, there are individuals within the community, whether previously targetted by the term or not, who still view queer as a slur, so while some find it acceptable to describe themselves as queer - ranging from the relatively "vanilla" sections such as homosexual, bisexual or transgender people, to non-binary people or individuals on the asexual spectrum - others find it personally offensive when used either as a personal descriptor - e.g. "hey, you're queer, right?" at a SGA. ultimately, it's similar to how black people have made steps to reclaim the n-word, previously used to oppress them, or how "lesbian" started off as a slur against women before being reclaimed by homosexual women.
all of this is to reach the ultimate conclusion that language and the terms we use are incredibly fluid, and constantly changing. back in the middle ages, "man" was used to describe people as a whole, of both sexes; not out of sexism, but because the gendered terms were different, werman and wiffman - "man" meaning person with the gender signifiers being "wer-" and "wif-" (it's actually where werewolf comes from! were- being man, -wolf being wolf!). to use other examples, in more recent decades the progression of gay from simply meaning happy to being synonymous with homosexuality, or even simply the jump from transsexual to transgender to better describe the status. or, to look at some more down-to-earth examples of terms arising that completely escaped the remits of their "inventors," the Red Pill analogy commonly used online by the anti-woke, anti-PC crowd famously comes from the Matrix, a movie made by the transgender Wachowski sisters, and which famously, by the reasoning of transgender critics and academics, seems to describe transgender experiences as they would be described by transgender people who hadn't realised that they were trans yet, as the Wachowski's - at that point, two brothers - were at the time.
All of this is to say, language and terms change all the time. to use an example from transgender culture, there's currently a move away from the terms Female-to-Male (FtM) and Male-to-Female (MtF) in favour of Assigned Male/Female At Birth (AMAB/AFAB), because they argue it's more fitting; they aren't going from female to male, they were always a man and they were simply raised to believe they were a woman because of a quirk of their biology (their idea). the term cisgender, as is almost acknowledged in this thread, is simply the antonym of transgender, which I feel like is being missed, but it, potentially, highlights the disconnect between trans-inclusionary and trans-exclusionary thinking. a trans-exclusionary person might use the terms "trans woman" and "woman," because they view trans women as separate to and different from "real" women, whereas a trans-inclusionary person, who would include trans women as women, would therefore describe the subset of women who are not trans as "cisgender women."
It's also potentially worth mentioning that trans-exclusionism in feminist spaces has a tendency to not only invite the far-right in but leads to so-called feminists working with groups that want to restrict abortion rights and women's rights generally, but I could never describe it as comprehensively as the Institue of Race Relations