Polyakov action: difference induced metric and dynamical metric
There are two manifolds that are involved in string propagation.
The spacetime in which the string propagates.
The worldsheet of the string itself.
The fields $X^\mu$ are embedding coordinates of the worldsheet in the spacetime manifold. This means that for each point $(\sigma^1, \sigma^1)$ on the worldsheet, $X^\mu(\sigma^1, \sigma^2)$ gives the coordinates of that point in the spacetime manifold.
In the case you are considering, the spacetime is taken to be Minkowski, so the metric is $\eta_{\mu\nu}$. Now we could ask
"Given that the worldsheet is a two dimensional embedded submanifold of Minkowski space, is there some way that this manifold inherits its metric from the metric on the ambient spacetime?"
This question is analogous to
"Given that the sphere $S^2$ is some two-dimensional embedded submanifold of Euclidean space $\mathbb R^3$, is there some natural sense in which it inherits its metric from $\mathbb R^3$?
The answer to both of these question is yes, and the metric on the submanifold that does this is precisely the induced metric. The formula expression the induced metric for a two-dimensional submanifold of some ambient manifold with metric $g_{\mu\nu}$ (not necessarily flat) in terms of embedding coordinates is $$ \gamma_{ab}(\sigma) = g_{\mu\nu}(X(\sigma))\partial_aX^\mu(\sigma)\partial_b X^\nu(\sigma), \qquad \sigma = (\sigma^2, \sigma^2) $$ You are right about the derivation of the induced metric, it comes from demanding that the distance measured between points on the embedded submanifold is calculated to be the same number whether you use the ambient metric, or the induced metric. To see that the above expression for the induced metric does this, simply note that the infinitesimal distance between any two points on the embedded submanifold can be written in terms of the ambient metric and the embedding coordinates as \begin{align} g_{\mu\nu}(X(\sigma))d(X^\mu(\sigma))d(X^\nu(\sigma)) &= g_{\mu\nu}(X(\sigma))\partial_a X^\mu(\sigma)\partial_bX^\nu(\sigma)d\sigma^ad\sigma^b \\ &= \gamma_{a b}(\sigma)d\sigma^ad\sigma^b \end{align} To get some intuition for all of this, recall that expression for embedding coordinates of $S^2$ in $\mathbb R^3$ is \begin{align} X(\theta, \phi) &= \sin\theta\cos\phi\\ Y(\theta, \phi) &= \sin\theta\sin\phi\\ Z(\theta, \phi) &= \cos\theta \end{align} and using these embeddings you should be able to show that the metric on the sphere is simply $$ \gamma_{ab}(\theta, \phi) = \mathrm{diag}(1, \sin^2\theta) $$
Let me know if that's unclear or if you need more detail!