Both can be done. Fine-Tuning the prompt is cheaper and can be done on consumer hardware. Fine-tuning the LLM model weights is expensive and needs cloud support.

Thanks for the reply, but when you mean "fine-tuning" for the prompt, do you mean fine-tuning of the LLM encoder of the prompt right ? (The thing that transforms the prompt into a sequence of embeddings?) But that is not cheap/easy to train ...

I know some systems also allow an extra fixed embedding parameter to the prompt, that can also be fine-tuned. But that is yet another thing that can be fine-tuned very cheaply.