Physical interpretation of $L_1$ and $L_2$ norms

While I can't satisfyingly answer your question about the $L_1$ norm, the $L_2$ norm has an easy answer.

Remember that a signal is generally, physically speaking, some sort of electronic oscillation in voltage (or current, if you prefer to think of it that way). It is well known that instantaneous power of an electronic signal is given by $$ \frac{dE}{dt} = R[I(t)]^2 = \frac{1}{R} [V(t)]^2 $$ Where $I$ refers to the current, $V$ refers to voltage, and the resistance $R$ is some time-independent coefficient. It follows then that the total energy expended, assuming a signal of finite energy is transmitted between times $\pm \infty$, is given by $$ E_{total} = \frac 1R \int_{-\infty}^\infty |V(t)|^2\,dt $$ The utility of the $L_2$ norm as being representative of "energy" does extend beyond this particular context. However, at the very least, it does make physical sense here.

Overall, the $L_2$ norm will always have a bit more mathematical utility than the $L_1$ norm since the $2$-norm is derived from an inner product, whereas the $L_1$ norm is not.

A useful property of the $L_1$ norm in the context of signal analysis is that an LTI-system is BIBO-stable iff its transfer function has a finite $L_1$ norm.