Wanted to update with something I found on Tesla forums:
When you insert the charger, push down on the handle gently in a way that the tip of the charger tilts up slightly inside the charge port. I don’t know if that’s normal but this is the only way my HPWC charges with green light.
I think it may have to do with the fact that the cable is heavy and it is naturally supposed to come down under its own weight and push the tip of the charger up. But since mine is wall mounted and the cable actually lays on the tool chest it doesn’t have any weight pulling the handle down.
Interesting. Looking at the pins inside the connector they are definitely designed to be straight and connect without any push or pull in any directions. I'm guessing something is not properly aligned in your charge port that requires this extra pull.