[This is now an answer to the edited question(s), with some details added.
My answer to the original question is kept at the very end.]
Firstly: The question is a good one, and it is not easy to find references on
this. I had spent too much time pondering about the failure of the double dual
argument (see below) before I finally heard the arguement given in the last
section below, indirectly from Fantechi, via Faber.
Assume $X$ is smooth projective.
Definition: An $S$-valued point in $M_I(X)$ is an $S$-flat coherent sheaf on
$S\times X$, with stable fibres of rank one, and with determinant line bundle
isomorphic to $\mathcal{O}_{S\times X}$, modulo isomorphism.
(I do not know if this is what Bridgeland meant, but to me this is resonably
standard.)
Comment: Stability for rank one means torsion free.
Existence: Let $M(X)$ be the (Simpson) moduli space for stable rank one
sheaves. Then $M_I(X)$ is the fibre over $\mathcal{O}_X$ for the determinant
map $M(X) \to \mathrm{Pic}(X)$. This map sends a sheaf $I$ (stable rank one
fibres) on $S\times X$ to the determinant line bundle $\det(I)$, and it is
trivial as a point in $\mathrm{Pic}(X)$ if it is of the form $p^*L$ with
$L\in\mathrm{Pic}(S)$. Then $I\otimes p^*L^{-1}$ is equivalent to $I$
in $M(X)(S)$, and it has trivial determinant. This shows that $M_I(X)$
indeed is a fibre of the determinant map.
Of course the determinant of an ideal $I_Y\subset \mathcal{O}_X$ is nontrivial
if $Y$ is a non principal divisor, so you cannot map such ideals to $M_I(X)$.
In any case, the ideal of a divisor, without the embedding, would only
remember the linear equivalence class.
For brevity, let $\mathrm{Hilb}(X)$ be the part of the Hilbert scheme
parametrizing subschemes $Y\subset X$ of codimension at least $2$.
Then there is a natural map $F: \mathrm{Hilb}(X) \to M(X)$ that sends an ideal
$I_Y\subset\mathcal{O}_{S\times X}$ to $I_Y$, forgetting the embedding. Since $Y$ is
flat, so is $I_Y$, and its fibres are torsion free (by flatness again) of rank
one. By the codimension assumption, the determinant of $I$ is trivial.
Theorem: $F$ is an isomorphism.
Comment: In the literature one sometimes finds the argument that if $I$ is a
rank one torsion free sheaf with trivial determinant, then $I$ embeds into its
double dual, which coincides with its determinant $\mathcal{O}_X$. This
establishes bijectivity on points. (For Hilbert schemes of points on surfaces
this is enough to conclude, since you can check independently that both
$\mathrm{Hilb}(X)$ and $M_I(X)$ are smooth, and that the induced map on tangent
spaces is an isomorphism.) I do not know how to make sense of this argument in
families.
Sketch proof of theorem: The essential point is to show that every $I$ in
$M_I(X)(S)$ has a canonical embedding into $\mathcal{O}_{S\times X}$ such that the
quotient is $S$-flat.
Let $U\subset S\times X$ be the open subset where $I$ is locally free. Its
complement has codimension at least $2$ in all fibres. By the trivial
determinant assumption, the restriction of $I$ to $U$ is trivial. By codimension $2$,
the trivialization extends to a map $I\to \mathcal{O}_{S\times X}$. This map is injective,
in fact injective in all fibres: The restriction to each
fibre $\{s\}\times X$ is nonzero (as $U$ intersects
all fibres) and hence an embedding ($I$ is torsion free in fibres). It
follows that the quotient is flat. There are some details to check, but
this is the main point, I think.
[End of new answer, here is the original one:]
If we attempt to define $M_I(X)(S)$ as the set of $S$-flat 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.
