I think that because 'woke' is an abstract noun, it has meaning in the head of the user. Yes, the original meaning was someone alive to injustice, but what that means in practice is open to interpretation. As with 'kind', or 'love' or 'beauty', people have different ideas of which actions qualify for the description.
As for 'white privilege', I think it is a very misleading term, as the word 'privilege' suggests that the phrase means that all white people (and their ancestors) have been in positions of power and superiority, which clearly is not the case. What it means is that white people have the advantage that whether rich or poor we don't have the added problem of being suspected of crimes, being overlooked for jobs, being unable to find products suitable for us (eg hair treatments for afro hair, or dark foundation creams) outside of cities or areas with large 'ethnic' populations, and so on. Some people of colour are rich, many white people are poor, but the things I've mentioned (and there are many others) only apply to people of colour.