As an example I think people from the American political left to somewhere(?) in the middle see it as what it has been introduced as, that being looking past the status quo and instead looking at your own values, i.e. the morality of homelessness and not having a disdain for them but empathy for them instead.
and then on the other side it feels like the people on the American political right see it as what this website describes it as “ A self-righteously moralistic person who behaves as if superior to others.”
I think the divide has originated from taking unlikeable behaviour and labeling that as ‘woke’ (in bad faith of course) and some people have just bonded to that definition so much that they see it as that.
At least that’s what I’ve noticed online over the past few (bonkers) years
The belief consists of two parts:
1. That truth is socially constructed thus when we see bad things, it means society created these bad things.
2. In order to determine what parts of society to cut-out to make society better, so bad things stop happening, use a critical theory to determine who should be removed from society so it can be more equitable (usually the stand in for good.
Woke normally holds that goodness is when results are equal, and if they are not equal, they have license to adjust them to equal (This is the core argument of Marxism, though woke could be said to be identity or social Marxism rather then just the economic Marxism presented, though in practice class identity was present from the start as well and expanded in practice under Mao).
There is no such thing as "society", just relationships between individual people. To get a better "society", you need people to act better. However, all of recorded history suggests that people are pretty universally willing to use other people as tools to benefit themselves. (Obviously not everyone does this all the time or to the same amount.) History also makes it clear that passing laws will not work: despite laws against things that are evenly timelessly non-virtuous, like stealing and murder, do not prevent murder and theft. In fact in Judeo-Christian thinking, to do this requires people receiving a "new heart, a heart of flesh instead of a heart of stone" from God. (I saw "Judeo-" because the passages is from Ezekiel, which is common to both. I do not know if rabbinical thinking agrees, however.) Even if it does not require a divine gift, certainly the problem has proven intractable up to the present time.
"determine who should be removed from society" is just a scary thought. Who gets to determine that? How can we be sure they are right? What prevents them from using this as a tool to eliminate people that are competitors or whom they simply dislike? In fact, this has a name: "to purge". The Soviet Union under Stalin and the Chinese Cultural Revolution were scary times.
I am extremely socially liberal, but have a very hard time aligning myself with the left because most members of that constituency seem completely incapable of recognizing this. They're so eager to repeat the errors of the leftist policies you list (along with other clearly non leftist examples like the Salem witch trials) that they're a danger to society.
They're zealots and need to be treated accordingly.