Thermostat with a servomechanism is self-aware of its state. A thermostat could be quite simply augmented to have a theory of another thermostat. Probably there are such coupled termostats.
I use intelligence mostly just as perspective of analysis to a system. E.g. Can it be seen to process information? How complicated is the transform from the information to some action? Does the processing lead to some good outcome, e.g. self-preservation of the system? Could the information processing be improved for this outcome?
Saying that a thermostat with a servomechanism is self-aware is just repeating the same broadening-beyond-usefulness with a different (though related) concept, in this case self-awareness. The difference between the self-awareness of humans and that which you see in a thermostat is vast, and understanding how a thermostat works gives no useful insight into how the human mind works.
Any system processes information, and in a thermostat it is not complicated at all. Even the most self-aware thermostats currently being manufactured are not going to improve themselves on their own volition. Applying the term 'intelligence' this broadly turns it into an empty tautology, or a little piece of uninformative noise on a communications channel, no more useful in understanding those systems than in understanding actual intelligence.
I use intelligence mostly just as perspective of analysis to a system. E.g. Can it be seen to process information? How complicated is the transform from the information to some action? Does the processing lead to some good outcome, e.g. self-preservation of the system? Could the information processing be improved for this outcome?