Music boxes weren't programmable in the sense that they allowed for a different algorithm to be performed. They changed the timing of certain operations.

>Or perhaps only things that can have branch instructions count

The author identifies programmability as the most interesting metric, which creates a "computer". Branching is necessary for that but not sufficient.

I think the most objective definition is probably Turing Completeness, but that's a pretty high bar.

I'd start the definition at "varied function and varied input data". Not necessarily branch instructions, the function can be much simpler.

So no, music boxes have no input data.

Music boxes can have input data. Drum with notches is input just as cards with holes is input. They just do not have the other part of varied function.

It might not be fast to replace the input data, but it is possible.

They can if you want them to! It's all semantics.

I personally wouldn't.