And does "woke culture" just mean a culture of being "aware of and actively attentive to important facts and issues (especially issues of racial and social justice)"?
It's similar to using words like "sheeple" to describe people who you disagree with. As you said, it's impossible to take you seriously when you use terms like that. That's what the outraged conservative commentators are latching on to. They're telling their listeners, "They think they are woke and you are asleep". Things are tribal. It's a powerful message.
But what if (and it's a big what if) there is a whole side of thought in America that believes issues of racial and social justice are not worthwhile and have already been settled? Would the term "woke" still be offensive then, based on what you just said?
I don't really know if the 2nd paragraph of your comment is supposed to be satire or serious, but that often happens when these undefinable words are thrown around as if they are cannon.