None of the siblings are right. The models themselves are idempotent: given the same context you will get the same activations. However the output distribution is sampled in a pseudorandom way by these chat tools. You can seed all the prngs in the system to always have reproducible output using sampling, or even go beyond that and just work with the raw probability distribution by hand.
Right. They are idempotent (making an API call doesn't cause a state change in the model[0] per se), but not necessarily deterministic (and less so as you raise the temp).
It is possible to architect things to be fully deterministic with an explicit seed for the pseudorandom aspects (which is mostly how Stable Diffusion works), but I haven't yet seen a Chatbot UI implementation that works that way.
[0] Except on a longer timeframe where the request may be incorporated into future training data.
