The core requirement for “programmability” is a machine whose behavior can be changed by altering symbolic instructions without rebuilding the machine.

The Banu Musa automatic flute player doesn’t meet this bar. The flute player was mechanically configured, not symbolically programmed. It had no conditional logic or flow control. No stored symbolic instructions.

To contrast, the Jacquard Loom used an externally stored instructions (punch cards). It allowed arbitrary-length instruction sequences, which is a primitive form of control flow.

The usual argument is that the Jacquard's loom inspired Herman Hollerith’s tabulating machine, used to tally the 1890 census, and became the basis of IBM.

Essentially, the speedups in textiles, inspired a speedup in computing (tabulating initially), which kicked-off the modern information technology industry.

And even if there wasn't any computation, it still is automated data processing (albeit simple).

> To those who are acquainted with the principles of the Jacquard loom, and who are also familiar with analytical formulæ, a general idea of the means by which the Engine executes its operations may be obtained without much difficulty. In the Exhibition of 1862 there were many splendid examples of such looms. It is known as a fact that the Jacquard loom is capable of {117} weaving any design which the imagination of man may conceive. It is also the constant practice for skilled artists to be employed by man­u­fac­turers in designing patterns. These patterns are then sent to a peculiar artist, who, by means of a certain machine, punches holes in a set of pasteboard cards in such a manner that when those cards are placed in a Jacquard loom, it will then weave upon its produce the exact pattern designed by the artist. 〈WEAVING FORMULÆ.〉 Now the man­u­fac­turer may use, for the warp and weft of his work, threads which are all of the same colour; let us suppose them to be unbleached or white threads. In this case the cloth will be woven all of one colour; but there will be a damask pattern upon it such as the artist designed. But the man­u­fac­turer might use the same cards, and put into the warp threads of any other colour. Every thread might even be of a different colour, or of a different shade of colour; but in all these cases the form of the pattern will be precisely the same—the colours only will differ. The analogy of the Analytical Engine with this well-known process is nearly perfect. The Analytical Engine consists of two parts:— 1st. The store in which all the variables to be operated upon, as well as all those quantities which have arisen from the result of other operations, are placed. 2nd. The mill into which the quantities about to be operated upon are always brought.

- Charles Babbage, Passages from the life of a philosopher.

I think you are implicitly invoking Turing completeness as opposed to “fixed-step recipes on a punchcard” as your requirement for something to be considered computing.

If a loom had a jump instruction, would you change your mind? (They did not.)

And yet we still have IDE defaults of 80 characters because of FORTRAN punchcards. That’s the heritage being invoked.

EDIT: I think all of these early devices also helped us to understand how to build multiplexers, which are the basic building block of any CPU. Given this instruction, I do a different thing.