I don't know that subjective experience corresponds to a "state". It may do.
Well, let's get back to that later.
...any hypothesis that doesn't explain the appearance of the star, in a seemingly random way, is obviously wrong. However, the internals could be mechanical, computerised, or some other kind of mechanism.
I agree, though I would phrase it a bit differently. The pattern we observe emerging suggests a pattern of behavior for the device. There's likely a tiny but real chance that it's coincidence, but we're probably ready to formulate a hypothesis about the machine's behavior--namely, that there exists within the machine some sort of mechanism influencing its behavior, and that this mechanism has a special property of excluding the star shape from the area of final arm movements when drawing the dot.
And as you said, it could be any of a number of things. I shouldn't be surprised when ripping the machine open to find a star shaped component. Neither would I be terribly disappointed upon opening it to discover that there wasn't one. But the one thing I would expect is that whatever the mechanism is, it is something that causes the arm to move randomly anywhere on the paper, except in the star shape. (I could easily be proven wrong).
So that's a rough model of science, and what we can determine using external behaviors. In this particular case, we can infer the existence of some sort of internal mechanism that directs the external behavior from the internal behavior (then again, it may not yet ruled out that the influence is an external mechanism either--we should probably check for eyes and stuff on the box).
However, in the case of the human mind, we can lift the lid off and see the workings - and we still don't know what produces the subjective experience. It might as well be a black box.
But it's all black box. The theory of gravity is virtually ancient, and yet we still have no clue what it is. We only see the arm move in particularly peculiar ways. We're still working on it.
One thing is for sure, though. You're allegedly describing the behavior. When we bump up to the meta level, something is causing you to describe your own behavior in this way. If we identify what is causing you to describe your behavior this way, we should automatically be including your subjective experience in our identification. If we aren't including it, then you must be failing to describe your subjective experience; or, alternately, you must be mistaken somehow about it.
There's no way I see for you to have a subjective experience, and to describe it, without the subjective experience playing a critical role in causing you to describe it that way. So if the robot researcher finds the cause for your describing it that way, and you do have subjective experiences, the robot researcher has found your subjective experience.