I think you need to follow your own advice! You and those like you don't have a monopoly on what this term means either. In fact I'd argue the people who created the term have the right to define it.
To your point, right leaning people can also be woke! I.e. concerned with inclusion, kindness and equity.
Perhaps it's time to retire the word. It's become some off the cuff, ad hominem insult by right wing leaning people rather than retaining any of the original meaning.
What I gather is that you actually have a problem with something specific that isn't related to "wokeness" but to the tribalism and "othering" that is going on among some left leaning people. This can be talked about specifically without abusing definitions or changing the meaning of words.
I'd go further and argue using woke the way it is used currently by right leaning people is a form of othering in and of itself.