b plays no role. It is a supernumerary parameter, or if you wish, it is redundant, and would result in regression instability in a spuriously underdefined system such that reparameterization is not optional, it is indicated.

The process is reparameterization, where, in the example, there is one supernumerary parameter that makes the system unnecessarily underdetermined. An alternative reparameterization is to $\frac{X_1}{X_1-\alpha X_2}$. The set $\{a,b\}$ has one more parameter than needed, but it is arbitrary to say which one, as it is the set that is overdefined, not $a$ or $b$ taken separately. Thus, $b$ by itself is only one of the two parameters that are problematic, and would likely not have a stand alone name. It is the set that is redundant, and would result in regression instability in a spuriously underdefined system such that reparameterization is not optional, it is indicated.