On a very fundamental level the LLM is a function from context to the next token but when you generate text there is a state as the context gets updated with what has been generated so far.
Its output is predicated upon its training data, not user defined prompts.
You're conflating UX and LLM.
Incorrect.
Prompts very obviously have influence on the output.
The LLM is also discrete.
its essentially a function that is called recursively on its result, no need to represent state