Expressing a set of 0-1 strings by Extended Formulation
An extended formulation (EF) of a polytope $P\subseteq \mathbb{R}^d$ is a linear system

\begin{equation}
\label{equation:ExtForm}
Ex + Fy  = g, y\geq 0
\end{equation}

in variables $(x,y)\in \mathbb{R}^{d+r}$ where $E,F$ are real
matrices with $d,r$ columns respectively and $g$ is a column vector
such that $x\in P$ if and only if there exists $y$ such that the equation above holds.

Thus the formulation you described in your question seems to be incorrect.
In special the right side of the inequality is expected to be a vector and not
a number.

I believe that the way in which you represent strings as extremal points of a
polytope is somewhat arbitrary. But a sort of canonical way is to represent
a string $x$ over $\{0,1\}^n$ by a vector $v\in \mathbb{R}^n$  such that
$v_i=x_i$.

However these extremal points are usually meant to represent the solutions to
a given combinatorial problem. For instance, in the cut polytope is the convex hull
the characteristic vectors of all cuts in the complete graph $K_n=(V_n,E_n)$:

\begin{equation}
\label{equation:CutPolytope}
CUT(n) = conv\{ \chi^{\delta(X)\in \mathbb{R}^{E_n}} | X\subseteq V_n\}
\end{equation}

where $\delta(X)$ is the set of edges from $X$ to its complement $\overline{X}$.
Similarly, the stable set polytope is the convex hull of the vectors corresponding
to the characteristic functions of all stable sets of $G$.

\begin{equation}
\label{equation:StableSet}
STAB(G) = conv\{\chi^S \in \mathbb{R}^V | S \mbox{ is a stable set of $G$}\}
\end{equation}

The TSP polytope $TSP(n)$ is the convex hull of all vectors corresponding to the characteristic functions of sets of edges representing the tours in $K_n$:

\begin{equation}
\label{equation:TSP}
TSP(n) = conv\{\chi^F \in \mathbb{R}^{E_n} | F\subseteq E_n \mbox{ is a tour of } K_n\}
\end{equation}

The interest on extended formulations stems from the fact that some polytopes,
such as the Permutahedron,which have an exponential number of facets admit extended formulations with a polynomial number of inequalities. However, recently Fiorini, Massar, Pokutta, Tiwary and De Wolf proved that any extended formulation of any of the three polytopes described above require an exponential number of innequalities.