It's simpler than that. Being "woke" means being self aware, to look at the world critically. For example, it means that when a politician tells you "the economy is better than ever", you look around and determine if your economic circumstances are better than they used to be. Maybe they're correct and you are an outlier, maybe they're using metrics that don't accurately reflect your life - you don't internalize what you're told blindly, you look at everything you're told critically

And it's not a big step from looking at the system critically to realize "hey, our system is absurdly stupid, on a fundamental level. It could definitely be changed for the better". That's not what being woke is - that's just a natural realization when you stop accepting what is and think about what could be

So the "war on woke" could be seen as a war on people who don't blindly accept what they're told. The term is mostly used correctly, most people just don't know what it actually means