'Female' and 'woman' are sex and gender respectively, right? A gender describes the traditional roles, expectations, and norms surrounding a sex. A sex describes one of two states of the body that are primarily distinguished by primary sexual characteristics and chromosomes. Do you agree with this?
Trans-identifying men, as you refer to them, wish to associate themselves with the female sex. In doing so, they recognise this means associating themselves with the roles etc. of females. Assuming we take it for granted that society does then proceed to associate them with those roles, expectations, and norms (most likely because they 'pass' as being female, or because they wear feminine clothing - women's clothing), they are by definition a woman.
That being the case, the remaining fundamental difference is sex. Recognising this, the term 'trans women' is most logical to describe what you term 'trans-identifying men'. It shows their sex does not correspond to the female sex, the roles etc. of which form the definition of woman.
Given that we have trans women now, it makes sense to distinguish trans women from non-trans women, i.e. females who associate themselves with the roles etc. of the female sex. 'Cisgender', where cis- means 'on this side of', was coined to mean 'not trans'.
I don't understand, given all of the above, how such a term can be seen as offensive. I am consigning you to a subset of your gender rather than your sex. Your model is the one that consigns people to a subset of their sex specifically trans people.
I don't think of myself as anyone important or even somewhat slightly self-righteous. I only came here to discover the views of Mumsnet users, who are generally known to be overwhemingly female and gender-critical. I wanted to compare them to the views of trans people (in general and transmedicalists or 'truscum', if you've heard of them) - simply out of personal interest. I cannot prove it to you but I genuinely did not know how hated and offensive 'cisgender' was seen.
This was always a benign thread. I'm not some secret trans activist trying to infiltrate your movement; I'm just someone who disagrees with you. Also note that I haven't even started university yet (I'm on my gap year at the moment); 'language we threw back at you years ago' makes no sense given my age.