Rule 110 can be specified with a rewrite system, also known as cellular automata:
https://arxiv.org/abs/0906.3248. Cellular automatons have a correspondence with contextual grammars:
https://www.cis.upenn.edu/~cis5110/notes/tcbook-lang.pdf. Each is equivalent to a Turing machine, another way of saying that there is a program for it which can be specified on a Turing machine with the usual Turing machine instruction set for writing, reading, and erasing binary digits on a tape. This usual program can then be "compiled" into a rewrite system corresponding to the instruction set for rule 110.
The reason rule 110 is said to be Turing complete is because someone went through the trouble of specifying an instruction set for rule 110 so that other people could verify that it would be possible to write programs with it. This is not the case for the people who claim that they are computers. They always leave the instruction set undefined which makes their claims hard to believe.
I personally have no problem with people who think they're computers but if they're not programmable then I'm not sure what the point would be of calling themselves computers.
