You ask:
So, if the AI is like that piano — generating probabilistic output without any internal motivation, can we even call it an 'agent'?
Of course, but the question is, is it the same sort of agency human beings possess to which the answer is no. What we call agency is responsible insofar as it is commensurate with the philosophical concept of agency. From WP:
Agency is the capacity of an actor to act in a given environment. In some contexts, the exercise of agency is linked to questions of moral responsibility... Agency may either be classified as unconscious, involuntary behavior, or purposeful, goal directed activity (intentional action). An agent typically has some sort of immediate awareness of their physical activity and the goals that the activity is aimed at realizing. In 'goal directed action' an agent implements a kind of direct control or guidance over their own behavior.
So, one simply has to pair up the comprehension to see what sort of alignment there is. An automatically playing piano fits the definition to a degree. It is autonomous in playing a song, but lacks moral responsibility. It's purposeful because its design embodies the purpose, and its awareness is limited to taking instructions from the humans given the input is given. Your example is a basic form of a machine or computer which takes information from the user, translates it into basic action, in this case generating a song encoded inside the device, and then making that song manifest. Control or guidance here might be understood as self-regulation of the mechanism for sound generation.
You ask:
So, if the AI is like that piano — generating probabilistic output without any internal motivation, can we even call it an 'agent'?
But yet, arguably it does have motivation, especially if we note that motivation simply means that which causes it to move. It doesn't have motivation like a concert pianist, but it certainly has a set of mechanisms to cause it to behave in a certain way mimicking a pianist. An automatic piano is motivated in a simpler, but not entirely dissimilar way. Our motivation may stem from the integration of the PFC atop the limbic system and brain stem, but the piano may have still have a simpler electronic control module.
You ask:
Does the fact that I (the human) provided the 'will' make the output mine? Or does the fact that the AI provided the 'form' make it the author, even if it’s a 'dead' author?
Causality is quite the tricky topic and there are many theories on what it is and how it "works". I personally subscribe to the belief that causality is a reflection of an information processing system's map of prediction of epistemically accessible events. That there are multiple models of what causes what that, in the vein of neopragmatism, serve the purpose and motivations of the language community. Of course, physical causality is an extremely important theory that is a product of scientific thinking. For Aristotle's causal schema, the material cause would be the physics of the machine, the formal cause would be the design of the piano, the efficient cause would be the means of manufacturing it, and the final cause would be the purpose it serves.
You say:
I press the 'Play' button (Efficient Cause). The piano mechanisms hit the strings and create the melody (Formal Cause). But the piano doesn't care about the music. It has no Final Cause (purpose/motivation).
On my reading, your analysis of the four causes seems to jumble up Aristotle's notions with your own. I would argue that you are the efficient cause, because you are the direct cause by way of 'play' to set the song in motion. The formal cause of the music is the device that has been design and built since it is responsible for the immediate production of the sound in a relatively autonomous fashion, and that the encoding of the song internally is the final cause because the whole purpose of the machine is to cause a song to be heard, and without which no song can be produced.
You ask:
Does the fact that I (the human) provided the 'will' make the output mine? Or does the fact that the AI provided the 'form' make it the author, even if it’s a 'dead' author?
So, let's apply the analysis to an LLM. The material causes are everything that describe the underlying computer the LLM "runs on". The formal cause would be the software that is trained and ready to respond to your question. The computer, software, and knowledge engineers would be the efficient causes in the LLM, and lastly the final cause would be the prompt you submit, as the output, what the system produces is a function of the input. Notice that none of this discussion has anything to do with will or will power.
I think you need to fundamentally reassess your understanding of the four causes, because your interpretation of the context seems to be off. LLMs are good search engines, but they are not debaters because they do not reason. They simply provide a probabilistically generated text which resembles reason. And if you can't reason, and they can't reason, you're likely to wind up with a heavily flawed dialectic unless you find a mechanism to self-correct.