Let’s try an experiment, hm.

“I am not a woman.”

Using the definition “a woman is someone who identifies as a woman,” would you call me a woman.

https://philosophy.stackexchange.com/questions/98474/is-this-a-fallacy-a-woman-is-an-adult-who-identifies-as-female-in-gender

There’s no real issue with recursive (self referential) arguments in philosophy or math. An example being the Fibonacci Sequence. I’m going to assume your criticism for this is that you, like many conservatives, think this definition is circular.

The following definition is not circular:

A woman is somebody who says they are a woman.

This definition proposes a test, “do they say they are a woman?”, to determine if somebody is a woman (according to the given definition). This test can be performed without needing to circularly apply the definition of the term “woman” ─ because we don’t need a definition of “woman” to know whether or not somebody says they are a woman.

You may argue it is not a useful definition, because it does not depend on what the person who says “I am a woman” means by the word “woman”, only that they use that word to describe themselves. Others will disagree. But the definition itself is not circular.

Perhaps it will help to make an analogy with a similar non-circular definition which was used historically, though is no longer used in modern times, but the definition was not contentious and I am not aware of anybody seriously arguing that the definition was invalid due to circularity.

Before marriages had legal status in modern law, it used to be that a husband and wife became married in a ceremony, in which a religious leader declared “I now pronounce you husband and wife”. This pronouncement itself used to be what made two people husband and wife, so if two people had not been married in such a ceremony where such a pronouncement was made, they would not be husband and wife.

So the definition of “husband” and “wife” included that the husband and wife had been pronounced as such, by the power vested in whoever officiated wedding ceremonies. (There were other aspects to the definition as well, but this criterion was required.) Does this mean that until modern times, marriages were meaningless, because being a “husband” or “wife” depended on a pronouncement being made, where the pronouncement itself necessarily included those terms which were defined by the pronouncement?

Of course not. This definition is likewise not circular, because we can apply the definition to determine if two people are husband and wife ─ i.e. has such a pronouncement been made by someone qualified to make it? ─ without having to more deeply investigate the meaning of the words in that pronouncement. The fact of the pronouncement being made, regardless of its meaning, is enough to satisfy the definition.