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There is a lot of Turing machine out there. Most of them are purpose-specific.
What make universal Turing machine universal? How do we know or prove if a computer is universal?
Edited: Is programmability of universal Turing machine the key to be universal?
You're confusing a few concepts.
"General-purpose computer" is a high-level, not very well defined term. It's used to refer to computer hardware that is not designed for a specific task and should be able to accommodate most of today's usual (consumer) workloads with reasonable performance and efficiency. Most commercially sold computers and smartphones are considered general-purpose. Examples for non-general purpose hardware would be hardware for database servers or a computer for genome sequencing with additional circuitry or FPGA's.
A universal TM is a TM $A$ whose language is
$$ L(A) = \{\langle M \rangle \# w \mid M \text{ accepts } w \} $$
This is just one of many equivalent ways to give the language. The point is that a universal TM can simulate any provided TM on any provided word.
If you "have" such a turing machine, it would be universal in the sense that you could use it to simulate any other turing machine. There isn't any meaning implied by the word "universal" other than that.
The term "universal computer" isn't used in computer science.
The turing machine is just a model for computability. It can be used to describe the process of computing anything that can be computed (cf. Church-Turing thesis). It's not an ideal model for computing by any other means.
Computer hardware is built to efficiently execute the widest variety of programs and therefore automatically strives to be like a universal turing machine. This, however, isn't really a goal per se. It's just a consequence of the power of the turing machine model.
