I think this narrow view on determinism comes up a lot. Essentially any program can be made deterministic over single inputs by fixing all side-inputs. But there is determinism over classes of inputs too, eg. an algorithm given input X deterministically outputs X+1.
I think the wider view is usually what people mean when they talk about nondeterminism in LLMs. Maybe because computers are traditionally so deterministic the wide view is almost taken for granted by programmers.