Feel free to move this thread to another section since this has turned into a how can I improve this image, which is great.
Soon after posting, I kept looking at the ice and knew that was a problem. I think this render is much better. The main differences are the material on the ice was scaled up and uses less displacement. I have a light focused on the ice to brighten it up a bit, but it was too bright and the refraction went nuts. I also switched the Reflections Override with a monotone studio light image.
I was already rendering this before I quickly glanced at that article you posted. His liquid settings are basically default minus the color thing with the neck. The most obvious thing Corona is not creating since the liquid is a hard line against the glass is the way liquid creeps up the side making additional refraction effects. In a still image, that can be faked in Photoshop obviously. My liquid actually has a little more refraction than his in the article. I rendered a test with just the liquid and it it most certainly refracting.
Appreciate the feedback though.