I think that a scientific/mathematical definition would work in this case - if you can simulate a turing machine on that particular machine, then it is considered a general purpose machine. Otherwise, its a special purpose machine.
But you can't actually simulate a TM on any machine with a fixed amount of memory. And any "special purpose" computing hardware would have at least a little memory. So I don't think that distinction works. Unless it only applies to machines with expandable memory? Technically, if it were hot swappable, the memory would be unbounded. But nowadays, the SD slot on phones is often under the battery, if it exists at all.
I mean, you can't simulate the weather exactly, but that doesn't stop us from writing "good" approximate simulations. It's not hard to simulate a TM, even if we can't build a real TM (unbound memory).
