If we attempt to define $M_I(X)(S)$ as the set of ideals $I_Z$ in
$\mathcal{O}_{S\times X}$, then that would not be functorial in $S$, as the 
inclusion $I_Z \subset \mathcal{O}_{S\times X}$ may not continue to be injective after
base change (in the counter example in the other answer, restriction to the problematic fibre gives the zero map). We could impose
"universal injectivity", but that is just another way of requiring the
quotient $\mathcal{O}_Z$ to be $S$-flat, so then we have (re)defined the Hilbert
scheme. 

Another common way of defining moduli of ideals is as the moduli space for rank
one stable sheaves (i.e. torsion free) with trivial determinant line bundle.
The resulting moduli space is isomorphic to the Hilbert scheme of subschemes of
codimension at least 2.

(I have not studied Bridgeland's paper, so I do not know the intended meaning there.)