Thanks for the detailed technical analysis, your conclusions are absolutely correct.
This is a known issue when using GPT-5.2 in Agent mode with a custom OpenAI API key. Cursor sends requests in the OpenAI Responses API format (which uses fields like reasoning, input, metadata), but the tools schema doesn’t match the standard format. It uses tools[i].name instead of tools[i].function.name, and it also sends custom tool types (for example type: "custom" for apply_patch) that aren’t supported by the standard OpenAI API.
The team is already working on this, but for now Agent mode with GPT-5.2 via BYOK OpenAI API isn’t properly supported.
Sorry for the inconvenience, we’re working on it.