Associated Legendre Polynomials and Legendre Polynomials are strongly connected. From what I read, Legendre polynomials is a special case of Associated Legendre polynomials where $m=0$. Thus the Orthogonality and parity condition followed from Associated Legendre Polynomial.
From alexjo's answer in this post Associated Legendre Polynomials!? , Associated Legendre polynomial was the solution with "a separation ansatz", and Legendre polynomial was the solution with azimuthal symmetry. Spherical Harmonic was then introduced with Associated Legendre polynomial multiply by a complex phase.
Is the above summary correct? or have I missed something?
From this post Proof that Legendre Polynomials are Complete, Legendre Polynomial itself had already formed a complete basis, so why do we need Associated Legendre Polynomial? Isn't it too excessive?
Further, I still have a hard time in dealing with the the geometrical picture of those functional basis. I think I saw somewhere Spherical Harmonic was represented with a vector on a sphere aiming at $m,(m-1),...,-m$. However, how could we visualize associated Legendre polynomials? Does those $m$ represented the similar thing?
Could you in help me to distinguish the usage of those function basis, please?