I get it now.
I really, really get it now.
After figuring out the stuff in LWJGL to make it work in a browser and show a black rectangle, I finally figured out how get a Slick2D app to do the same thing.
So now, from exactly the same code base, I can generate both the stand alone runnable jar and the web browser app.
The deployment is different, but that’s fine.
I can do it.
Yay for me.