What do you mean with "stateless" here, and which of those meet the stateless criteria? I have a hard time imagining any of those without state.

Deployments of server applications do usually involve some inherent state, like the contents of a database or the fact of which services are running. In that respect, some deployment tools in the Nix ecosystem are not isomorphic. But the software management and package management components, for example, are declarative without involving any reasoning about the current state. It is in that sense that those components are 'stateless' in a way that the rule of thumb proposed in the GP is not.

This is an option, but it is certainly not a requirement. If I say "I want cheeseburger", it is irrelevant to me if the cheeseburger is made from scratch or had been partially assembled ahead of time.

You are concerning yourself solely with configuration that can be generated not only declaratively, but also fully deterministically, but configuration systems can indeed exist where the environment isn't sufficiently deterministic. In such cases, taking the current state of the world, the user's request, and then determining the appropriate (and preferably minimal) actions to take to harmonize those without having to have them specified by the user is a declarative approach.

For example, I can go to the AWS console and turn up a job with a set of parameters, or I can have a configuration file and a daemon and if

    {
        cpu: 5,
        mem: 20,
        image: "my_docker_url",
     }