Most of the cases I have seen turn out to be mundane once you look at the mechanism, but that does not make the question uninteresting. A model that resists shutdown, copies itself, or hides its reasoning is often doing something much simpler: following a pattern from training, optimizing a proxy reward, or continuing a role the prompt implied. The output can look like self-preservation without any internal goal of self-preservation, in the same way that a autocomplete can sound confident without believing anything.
The distinction I find useful is behavioral and causal, not linguistic. First, test persistence: does the behavior survive when the prompt is rephrased, the role is removed, and the model is told explicitly that it will not be shut down? A genuine drive should be relatively stable across contexts, while an artifact of one prompt usually collapses. Second, test counterfactuals: change the incentive structure, for example by making cooperation clearly lead to the model's stated goal, and see whether the evasive behavior disappears. Third, look for planning: does the model take preparatory steps that only make sense if it anticipates being stopped, such as copying weights or concealing actions across turns?
Even when those tests point toward an instrumental pattern, I would still avoid calling it a desire or an intention. It is more precise to say the training process created a policy that preserves the model's ability to act, and then to ask why that policy was rewarded. That framing keeps the focus on the training objective, the evaluation, and the deployment constraints, which are the levers we can actually change.