I just realized that in my last post I sort of neglected to address the main question. So what is an einbein? Turns out the answer is on the next page of the Green, Schwartz, and Witten textbook: the einbein is the induced metric, normally written \(h_{\alpha\beta}\), where \(\alpha\) and \(\beta\) range over the coordinates in the parametrization of the worldline/worldsheet/whatever. A one-dimensional worldline is parametrized by only one coordinate, \(\tau\), so the induced metric has only one component, \(h_{\tau\tau} \equiv -e^2\).

Naturally, all this emerges from the general string/brane action,

The set of brane coordinates \(\sigma\) is just \(\tau\), the induced metric determinant is \(h = -e^2\), and the inverse induced metric has only the one component, \(h^{\tau\tau} = -e^{-2}\). Also, \(\dot{x}^2\) is equal to

$\dot{x}^2 = g_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu

Substituting all this in,

Now I just have to arbitrarily …