Every time I go to chatgpt.com I type chat into the address bar and it autocompletes to the correct site. Also when talking to my coworkers it's taken the same status as "googling" something. We just call it "chat". So yes, I also feel it's odd to see chat.com go to chatgpt but it's the shorthand some people are already using.
Examples of what my coworkers and I say at the office:
"What does chat have to say about x?"
"Did you ask chat"
"Did chat find the answer?"
"Yeah, chat scripted that one."
Yeah, me too. And if you squint your eyes and tilt your head you can kind of make the connection. Any large Language Model is made of the "chat" from billions of messages floating around on the internet.
So when you ask chat a question you are in reality asking an algorithmic simulation of mass internet communication.
In my mind the connection works, which is convenient with the new chat.com domain and all...
Kind of funny, because there's another product we use that's just called Chat, so we'd never say that. "GPT" is a better shorthand IMO. It's also not specific but I don't know of any other product that uses it in the branding.
Examples of what my coworkers and I say at the office: