The word "woke" was first used in the 1930s, and critical theory began its intellectual popularity in the late 1800s. Marx and Clausewitz were early adopters of critical theory, but Engels, Gramsci, Hegel, and many others who popularized the movement did their work in the late 1800s and early 1900s. In other words, critical theory predates "woke" by a century.

The word "woke" only attached to critical theory long after both had come about.