Its pretty clear from Anthropics own Console what info models have. Any non-API 3rd party tools have system prompts which has impact on responses.
I have posted tests of various models in many threads here and will leave it at that, this has been officially addressed by Cursor Team members in several threads as well.
If you have clear info there is an issue with any model please create a full Bug Report with a Request ID that has privacy disabled so the Cursor Team can review the process and what happened on that specific request. Please tag me there.
Hallucinations of a model claiming to be another model are not proof, it should be something more substantial.