Originally, actually there was a relationship between Cursor & OpenAI. Something like Cursor was supported by the OpenAI startup fund. So Cursor seems to have branched out. I think they are just emphasizing the models they find most effective. I'm surprised they haven't (apparently) incorporated Claude prompt caching yet for Sonnet.
Try using Gemini thinking with Cursor. It barely works. Cmd-k outputs the thinking into the code. Its unusable in chat because the formatting sucks.
Is there some relationship between Cursor and Anthropic, i wonder. Plenty of other platforms seem very eager to give users model flexibility, but Cursor seems to be lacking.
I could be wrong, just an observation.