I believe the useful term my logic professor used was axiom schemata. For instance, What Hofstadter calls Axiom 2 says
$Mx\to Mxx$ for all non-empty strings $x$
This is not a formal axiom of the system because $x$ is not a symbol in the system, nor is there a universal quantifier in the system. (Let's assume for the sake of argument that $\to$ was a symbol in the system. I'll return to this point.) It is a placeholder for the infinite number of statements
$MI\to MII$
$MU\to MUU$
$MII\to MIIII$
$MIU\to MIUIU$
...
So it's an axiom expressed in the metalanguage of the system. This really should be formally distinguished from the pure axioms of the system (like $MI$ is), because an infinite number of axioms are obviously harder to work with.
That said, $\to$ isn't actually a symbol in the $MU$-system. What Axiom 2 really says (again, using traditional logic syntax) is
From $\vdash Mx$ to infer $\vdash Mxx$ for all non-empty strings $x$
This is not expressible in the language of the system for a whole new reason: the language has no "awareness" of what its theorems are. I think it's typical and reasonable to say that rules of this sort would be formally distinguished from axioms (which are well-formed formulas that we give special meta-meaning to).
You have a couple of nigh-equivalent ways of formulating the system that will all be equally useful in "solving" the system. You can say, as Wikipedia seems to, that there is one axiom and four rules of inference of the system. Or you can do something more along your lines, which is to add $\to$ to the language, have one axiom and four axiom schemata, and introduce Modus Ponens as your sole rule of inference. The choice is completely stylistic, as long as you're more formal than Hofstadter was on pages 33-35 of the text. ^_^