If I can program it to execute a sequence of arithmetic and logical operations that approximate a Turing machine (with a finite band), and reprogram it at a lat

If I can program it to execute a sequence of arithmetic and logical operations that approximate a Turing machine (with a finite band), and reprogram it at a later date to execute a different sequence of such operations, that's a computer to me. I wouldn't count ASICs, but the PIC12F508 or the 3-cent microcontroller referenced in the post definitely count. Though by my definition of requiring reprogrammability and Turing completeness I am purposefully excluding many things that have historically been considered computers, like the many mechanical computers of the 19th and 20th century. From that standpoint I can see how some people might count ASICs as computers, even if I don't think that fits modern usage.