By and large, the white people who strive to be anti-racist recognize the concept of cultural appropriation and therefore if they are aware that the term "woke" is AAVE (African American Vernacular English) and belongs to black culture, they will not apply it to themselves.
If they are not aware of this, they are still not likely to apply it to themselves because they see it as a self-flattering term (like "ally").
Whereas the anti-woke are self-proclaimed as such, and they use the term "woke" primarily to refer to anti-racist white people, which figures because people who self-describe as anti-woke are also not likely to give a damn about cultural appropriation.
Black people on the other hand continue to use the word "woke" (it does, after all, belong to them) to refer to the state of awareness of one's own racist oppression, as exemplified in Keith Knight's series "Woke."
So anti-woke people have defined themselves in opposition to something that largely does not exist-- self-identifying woke white people.
What exists are anti-racist white people, but anti-woke people won't acknowledge this because it makes it even more explicit that they oppose anti-racism. Which would strongly imply being pro-racism.
What also exist are black people who do identify themselves and each other as woke, but anti-woke people also won't acknowledge this, or at least in the sense that black people are using it themselves, because again, anti-woke folk are not exactly eager to identify themselves as opposing becoming aware of your own racist oppression.
(None of which is to say that people always fight racism productively. Sometimes they screw up. Sometimes badly. Sometimes they're actually bad people. These people should be called out. But you don't call them out by being "anti-woke," and you're in a terrible position to call them out if you don't do jack to fight racism yourself, regardless of how you self-identify.)