= Solution
The <canonical momentum> equation is $\dot X=eP+uX'$. Eliminating $P$ from the <phase-space action> yields the <Lagrangian density>
$$
\mathcal L=\frac{(\dot X-uX')^2}{2e}-\frac{eT^2}{2}X'^2.
$$
This is the <Polyakov action> after identifying its inverse <metric tensor> density as
$$
\sqrt{-\gamma}\,\gamma^{\mu\nu}
=\frac1{Te}\begin{pmatrix}-1&u\\u&T^2e^2-u^2\end{pmatrix},\qquad
\gamma=\det\gamma_{\mu\nu}.
$$
The <determinant> of this <matrix> is $-1$, as required in two dimensions. The <Weyl transformation> leaves the <matrix> unchanged, so $e,u$ encode the metric modulo its <conformal factor>.
Define the <induced worldsheet metric> by $g_{\mu\nu}=\eta_{mn}\partial_\mu X^m\partial_\nu X^n$. Varying the independent inverse <metric tensor> in the <Polyakov action> gives
$$
g_{\mu\nu}-\tfrac12\gamma_{\mu\nu}\gamma^{\rho\sigma}g_{\rho\sigma}=0.
$$
Consequently $g_{\mu\nu}=f\gamma_{\mu\nu}$, where $f=\tfrac12\gamma^{\rho\sigma}g_{\rho\sigma}$. For a nondegenerate Lorentzian <string worldsheet> with the usual compatible time orientation, take $f>0$, or $\gamma_{\mu\nu}=e^{2\omega}g_{\mu\nu}$. The undetermined function is precisely <Weyl invariance>. Since $\sqrt{-\gamma}\,\gamma^{\mu\nu}g_{\mu\nu}=2\sqrt{-g}$, elimination of the independent <metric tensor> gives
$$
\boxed{I_{\mathrm{NG}}=-T\int dt\,d\sigma\,\sqrt{-\det g_{\mu\nu}}.}
$$
Thus the <Nambu–Goto action> measures minus <string tension> times Lorentzian <worldsheet area>. The metric elimination holds in the nondegenerate interior; null endpoints are understood as boundary limits.
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