Do you really think the Black Lives Matter organization speaks for the whole of the African American community?

You can't focus on the speaker. You have to focus on the hearer. When cities pave their streets with "BLACK LIVES MATTER," and Black activists ask every White pol they can find whether "Black lives matter," and reject "All lives matter" as racist, and others natter on about White privilege and the Kafkaesque White fragility, White people feel threatened. If we are trying to figure why "woke" got hijacked, it makes absolutely no difference who said what or who spoke for whom or who was reasonably afraid of what. All that matters is that the word is seen by Whites as expanding its reach to systemic issues. And that perception, if not the assessment of threat it poses, seems to me accurate.

I'm not sure the anti-woke mob is as irrational as you say. You may be right that they have no reason to fear woke Black people who only want equality but that's not the "woke" they are anti-. You can say what "woke" means when you use it, and you say what it means when people you support use it, but you cannot say what the "anti-woke mob" is against. They are against what they perceive to be a totalitarian war on bad thoughts. They are against reparations from their pockets. They are against speakers being canceled, whether or not race is their thing. They are against trans women in sports. They are not using "woke" to mean "You're a Black person in a White place, so watch your back." But you can't dismiss their fears by critiquing their word choice.