"Taking people’s words out of context to make it sound like they said something bad as a joke is by no means uncommon in comedy."

Adults are responsible for what comes out of their mouths, or keyboards. Calling libel "just a joke" doesn't make it not libel. It's not far removed from "it was just a prank, bro".

"To me, this reads as a vanity case. JK Rowling got her feelings hurt"

First of all, no it isn't just about hurt feelings, it's about intentional misrepresentation. It is valid to object to being misrepresented whether or not you personally found it hurtful.

Secondly, I want you to read your post back to yourself, aloud, and realize the entire backlash against Rowling from trans people is because she hurt their feelings. By your own logic, that's also a vanity case.