I'm not sure you quite have the meaning of "critical". Horkheimer wrote an essay called Traditional And Critical Theory. I think that is close to an origin point for this. Sure it kind of assumes a Marxist view of the world, but lacking from what you said is that for it to be "critical" it must do three things.This is a pretty good start.
Critical in this sense should be taken to mean operating from the perspective that knowledge is socially constructed and education is a political project embedded within a network of social institutions that reproduce inequality. I'm happy to take social justiceWP in the usual sense.
1. describe the world (that is the bit you seem to be describing).
2. it has to be practical, the problem with traditional theory is it's requirements for evidence stand in the way of answering questions that we want to answer. Neo-marxist theory is substituted for evidence.
3. it has to be normative, it has to be a moral call to action.
In some ways, I don't see the difference between critical theory and critical social justice.
That is a definition. I guess you could also call it all academic disciplines that supress results that go against the woke world view. The resignation of Jonathan Haidt has been mentioned I think.Can we agree that "woke" academia covers all the different academic disciplines which consciously take a critical stance, e.g. multicultural education, anti-racist education, postcolonialism, intersectionalism, gender studies, critical race theory, etc.?
Sure, rhetorically it is foolish to call it that when talking to the public. It makes it clear that you aren't just talking about some kind of naïve programme to treat people fairly and get's in the way of motte and bailey equivocation. It's like how you might refer favourably to some theory as Marxist in academic circles, but you probably wouldn't do that on CNN. Were I woke, I would certainly not call it that.If you were to google "Critical Social Justice" as a phrase, you'd mostly get hits from anti-woke people in the Pluckrose camp. Seems to me the phrase is more often used to characterize one's opposition than one's allies.