# What are different definitions of Universal Turing-machine?

I am not sure what is the appropriate way for this, but I would like to collect different possible definitions/variants of Universal Turing-machines. Here are the ones I know, post below if you know other definitions/comments/don't agree. For simplicity, I assume that we talk about two-tape machines with alphabet {0,1,*}, but let me know if in some definition this might make a difference. I denote by T(x,..,z) the output of Turing-machine T when the inputs on its tapes are x,..,z, if it halts, and "running forever" otherwise.

Exists. U is universal if for every T there is a p such that for every x we have U(p,x)=T(x).

Computable. U is universal if there is a computable encoding E of Turing-machines (labeled directed graphs) into strings such that for every T and x we have U(E(T),x)=T(x).

Fix. Fix a (computable) encoding E of Turing-machines into strings. U is universal if for every T and x we have U(E(T),x)=T(x).

Strict. In both the Computable and the Fix definition, we can also demand that for every input that is not of the above format, U does nothing useful (halts without output, outputs the empty string, or runs forever).