I know that physicists call $\ell$ and $m$ the "azimuthal" and "magnetic" quantum numbers, respectively. But those sound very physics-y. (I am actually a physicist, but still.) Are there names for these considering the spherical harmonics simply from a mathematical perspective? Maybe "zonal" and "modal" numbers? Any precedent for those???
So far as I'm aware, the names for those indices are inherited from the names for the corresponding arguments of the associated Legendre polynomials that show up in the definition of the spherical harmonics; thus, in $Y_\ell^m(\theta,\varphi)$, $\ell$ is the degree, and $m$ is the order.
But if you're communicating with physicists or chemists, I would recommend just terming them the quantum numbers, since those names are more intuitive anyway...