For what I know, the definition of a universal Turing machine is something along the lines of the following (of course, details might vary from source to source):
A Turing machine $M$ is called universal if there are computable functions $f:\Bbb N^2\rightarrow\Sigma^*$ and $g:\Sigma^*\rightarrow\Bbb N$ such that for any $e,n\in\Bbb N$, $\varphi_e(n)$ is equal to $g$ applied to output of $M$ on input $f(e,n)$ (here I use (in non-standard way) $\Sigma^*$ to denote possible tape contents with finitely many non-blanks).
(basicaly the idea is that we always can encode the input, let $M$ run on it and decode output so that we compute any computable function)
However, there are many machines which are "weakly universal", for example machines which simulate rule 110. My question is, what does that mean? These notions are usually understood to be intuitive by community, but I have never managed to find a (semi)formal definition of universality. Indeed, most sources that call machines like Wolfram's (2,5)-TM (weakly) universal define universality as "able to simulate any Turing machine", whatever that means. I have decided to make my question specific as follows:
I suspect this might have been asked before, but I couldn't find anything by searching.
Thanks in advance for all the answers.