At the very least, if the people who once used the word "woke" to describe their endorsement of certain illiberal beliefs don't want to be called "woke", they need to come up with a term that everyone else can use.

I understand the rhetorical advantage of not being able to be pinned down--it's right in line with a kind of postmodern power-play. But then they can't very well complain if others seize the discourse, or simply add enough clarity to know what is being talked about, by applying a label.