TL;DR
It's a gene called the SRY gene. If its present, and it only exists on the Y chromosome, the off-spring is male. If it's absent the off-spring is always female. This is why the "trans" movement claims "women" can have an XY chromosome as they can but the sex determining SRY is absent thus the default state is achieved, female.

Long form explanation
https://medlineplus.gov/genetics/gene/sry/
The gene responsible for determining the male sex is called the SRY gene, which stands for "sex-determining region Y." This gene is located on the Y chromosome. In the XY sex determination system, which is found in humans and most placental mammals, the genetic sex of an offspring is determined by a pair of sex chromosomes: an X chromosome from the mother and either an X or Y chromosome from the father. If the offspring inherits an X chromosome from the father, it will be genetically female (XX), while if it inherits a Y chromosome from the father, it will be genetically male (XY).

The SRY gene on the Y chromosome initiates the release of testosterone and the formation of male sex organs. SRY is a transcription factor that can turn on the expression of other genes, serving as a master switch to turn on the suite of "male" genes in a developing organism. In the absence of the SRY gene, the default developmental pathway is female.

The presence of the SRY gene on the Y chromosome is the key factor in determining the male sex in species with the XY sex determination system, such as humans.


Edit to add:
Thus to answer the OP's question, "What is a woman", it is simply a person that does not have the SRY gene present.

Here is a great video from 10 years ago that explains the whole SRY gene. it also explains why men have nipples and why there's a "joint" line on the scrotum (lol yeah, that was a new one for me)
And while it doesn't explicitly state why some women can have an XY chromosome it can be inferred as they do talk about the SRY gene turning on, or activating, and thus starting the development of male genitalia.