What does the degree and order of Spherical Harmonics mean?
It's Legendre polynomials, not Lagrangian.

The zonal harmonics (the $Y^0_l$ terms) depend only on latitude. These zonal harmonics are closely associated with the Legendre polynomials $P_l(x)$, where $x$ is the sine of the geocentric latitude.

The tesseral and sectoral harmonics (the $Y^m_l$ terms, where $m\ne 0$) are closely associated with the associated Legendre functions $P_{lm}(x)$.  (Note: These are sometimes called associated Legendre polynomials, but they aren't polynomials.)

Spherical harmonics are widely used in physics, so the presentations readily found on the internet generally reflect how physicists use spherical harmonics. There are some key differences between these easily found references on the internet and the forms used to represent gravitation. One is that physicists typically use colatitude, and hence use $P_l(\cos\theta)$ as opposed to $P_l(\sin\phi)$. Another is that physicists tend to use the complex form, while gravity models are expressed in terms of real sine and cosine forms. Finally, a set of gravity coefficients is inevitably going to be fully normalized coefficients, while physicists use a form with a different normalization.