You are misunderstanding me.

Sure, perhaps that is definition used by those who adopt the term.

The problem is that the culture outside this small group has agreed that the term refers to people who are insufferable and holier-than-thou.

I know people who care about these sorts of things. They talk about social inequality. They vote for politicians who want to reform the system. They work in nonprofits to make the world a better place. They attend conferences to learn about the newest discoveries in intersectional studies.

None of them call themselves “woke”, or use the word at all. Why? Well because the word and its cultural associations are cringy. They want to be seen as sensible, reasonable people who have realistic solutions to real problems - using the word “woke” would undermine this goal. It doesn’t matter what the actual, literal definition of the word is - simply using it would undermine their credibility, since it would indicate a lack of social acuity. It’s like someone using the term “retarded” to refer to someone with a developmental disability. Even if they use the word in a completely kind and clinically appropriate way, their general credibility is undermined because polite society has determined that it is an inappropriate word to use.