I mean you basically don't have a problem with the dictionary definition
I have a problem with it in the sense that it means a lot more than anti-racist. Perhaps the term originated in that context, but it certainly has evolved into something more.

Like, if you think black people should get an advantage today to compensate for historical abuses from centuries past, this is unfair imo.
Excellent. I don't think you're woke. It's not something left-leaning people automatically become. For example, you're not typically leftist when it comes to trans rights. You're more balanced and try to see it from both sides of the debate.

How about we agree that you're not using the word "woke" correctly according to the dictionary and leave it at that.
No. You're using an outdated definition. Try the wikipedia page...

https://en.wikipedia.org/wiki/Woke

Woke (/ˈwoʊk/ WOHK) is a term, originating in the United States, that originally referred to awareness about racial prejudice and discrimination. It subsequently came to encompass an awareness of other issues of social inequality, for instance regarding gender and sexual orientation. Since the late 2010s, it has also been used as a general term for left-wing political movements and perspectives which emphasise the identity politics of people of color, LGBT people, and women.
With this in mind, it's a term that is basically used for people who are neck deep in identity politics.

lol, so now the fact checkers are in on the conspiracy.
Yes.

So the BBC, that refuses to report anything negative about the Tories, is more left-wing than the Guardian and the Times, who both do.
The BBC is very strange. On the one hand, they promote identity politics with a left wing agenda. On the other, they are pro-Tory. But they are pro-Tory because they are state media. They will be pro whoever is in power.