Neither option implies you own the model or don't: OpenAI owns the model and uses prompt tuning for their website interface, which is why it changes more often than the underlying models themselves. They also let you fine tune their older models, which you don't own.
You also seem to be missing that in this context prompt tuning and fine tuning are both about downstream tasks where the "user" is not you as an individual who's fine tuning and improve prompts, but the people (plural) who are using the now improved outputs.
These aren't the contexts that invite the scepticism though (except when the prompt is revealed after blowing up Sydney-style!)
The "NN provided incorrect answer to simple puzzle; experts defend the proposition the model has excellent high-level reasoning ability by arguing user is 'not good at prompting'" context is, which (amid more legitimate gripes about whether the right model is being used) is what is happening in this thread.
Neither option implies you own the model or don't: OpenAI owns the model and uses prompt tuning for their website interface, which is why it changes more often than the underlying models themselves. They also let you fine tune their older models, which you don't own.
You also seem to be missing that in this context prompt tuning and fine tuning are both about downstream tasks where the "user" is not you as an individual who's fine tuning and improve prompts, but the people (plural) who are using the now improved outputs.