I don't think any politician is actually disagreeing about what a woman is. Starmer has been vilified in the press, but Jacob Reese Mogg in a radio interview said basically the same thing as Starmer does, and possibly the only thing they can really say.
The law can be read here: www.legislation.gov.uk/ukpga/2004/7/contents
The law differentiates between a natal woman and a trans woman and determines when the two are treated equally under law and when they are not. The job of the law is to balance the rights and needs of the different groups and it tries to do so quite well.
This idea that is being spread, that some politicians do not know what a woman is, is false. Certainly at the level of politicians like Starmer with his legal background he is well aware of when the law treats a trans woman as a woman and when it doesn't and any debate on changing the balance of the law needs to go into a lot more detail and depth than the really shallow sound-bites we're seeing in the popular press.
Both sides have needs and rights, and there is a lot of detail there which I believe is missed in the sensationalist reporting we see. The job of the press includes to educate and inform. Sadly our popular press does neither.