I have been looking at the Laplace equation $\nabla^2 f = 0$ in various dimensions. In 3 dimensions, the angular equation leads to the well-known spherical harmonics, defined up to normalisation as \begin{align} Y_{lm}(\theta, \phi) = P^{|m|}_l(\cos\theta)e^{\imath n\phi}. \end{align} Here, the $P^{|m|}_l$ are the associated Legendre polynomials, defined by \begin{align} P^{|m|}_l(t) = (1-t^2)^{\frac{|m|}{2}}\frac{d^{|m|}}{dt^{|m|}}P_l(t), \end{align} where the $P_l$ are the Legendre polynomials.
This wikipedia article gives the Legendre polynomials by the Rodrigues formula as \begin{align} P_l(t) = \frac{1}{2^ll!}\frac{d^l}{dt^l}(t^2-1)^l. \end{align}
This paper I found, which discusses the Laplace equation in $p$ dimensions, gives the Legendre polynomials in $p$ dimensions by the Rodrigues formula as (equation 4.31 on page 69) \begin{align} P_l(t) = \frac{(-1)^l}{2^l(l+\frac{p-3}{2})_l}(1-t^2)^{\frac{3-p}{2}}\frac{d^l}{dt^l}(1-t^2)^{l-\frac{p-3}{2}}. \end{align}
$(l+\frac{p-3}{2})_l$ denotes the truncated factorial, i.e. $(n)_m = n(n-1)\cdot\cdot\cdot(n-m+1)$. For the case of $p = 3$, this is identical to the other formula, so I then assume that the wikipedia formula is implicitly for the 3-dimensional case (is this correct?).
I should also mention that I am a physicist, so I am familiar with a lot of the geometric reasoning in the case of 3 dimensions particularly, and the way I ask my questions may reflect this. Now this is what I am unsure about:
I have been through the paper I linked, and the $p$-dimensional case derivation makes sense to me. In this case, the argument $t$ of the polynomial is then interpreted as the cosine of the angle between points on the unit sphere, and we typically choose one of those points to be fixed as the 'North Pole'. So it makes sense to me how the spherical harmonics are associated with the sphere of a certain dimensionality.
However, the Legendre polynomials themselves seem '1-dimensional' to me, as in they take a single argument $t\in[-1, 1]$. Furthermore, there are alternative derivations of the Legendre polynomials that have nothing to do with Laplace's equation, and its associated dimensionality. For example, at the top of the wikipedia article linked above, there are 3 different definitions which I cannot see how to relate to some 'dimension'.
So my question is really what is the dimension of a Legendre polynomial?
If my question is unclear, or perhaps ill-defined, I am trying to understand why wikipedia says one thing, which when I compare to the $p$-dimensional case, is the $p=3$ case, yet there is no discussion of dimension on the wikipedia page - I cannot see where this additional information comes from.