Let $\mathbb{R}^{2,d}$ be the $(d+2)$-dimensional space with semi-Riemannian metric $\eta$ with signature $(-,-,+,\dots,+)$. We shall enumerate the Cartesian coordinates on this space as $X = (X^0,X^1,\dots, X^d,X^{d+1})$. Let us define the following map $\varphi:\mathbb{R}^{1,d-1}\to \mathbb{R}^{2,d}$: $$\varphi(x)=\left(x^0,\dots, x^{d-1},\frac{1-x^2}{2},\frac{1+x^2}{2}\right)\tag{1}$$ where $x^2$ is the Lorentzian norm squared of $x$. This is an embedding of $d$-dimensional Minkowski space on $\mathbb{R}^{2,d}$, in which it is realized as a section of the lightcone, such that the pullback of the metric $\eta$ in $\mathbb{R}^{2,d}$ by the embedding coincides with the Minkowski metric in $\mathbb{R}^{1,d-1}$. Now, we observe that if we define the following function on $\mathbb{R}^{1,d-1}$:
$$\sigma(x,y)=-2\eta(\varphi(x),\varphi(y))\tag{2}$$
then an explicit computation reveals $\sigma(x,y)=(x-y)^2$. It turns out $\sigma(x,y)$ is the geodesic distance between $x$ and $y$ in $\mathbb{R}^{1,d-1}$ (maybe a more precise terminology in Lorentzian signature is that it is the Synge world function).
On the other hand, let us define this other map $\psi:\mathbb{R}\times S^{d-1}\to \mathbb{R}^{2,d}$ given by $$\psi(\tau,\Omega)=(\sin\tau,\Omega^1,\dots, \Omega^d,\cos\tau)\tag{3}.$$
This map is one embedding of the Lorentzian cylinder with metric $g$ given by $$g=-d\tau^2+\gamma_{S^{d-1}}\tag{4}$$
where $\gamma_{S^{d-1}}$ is the round metric on $S^{d-1}$. It again realizes that space as a section of the lightcone. Like in the previous case, the pullback of the metric $\eta$ on $\mathbb{R}^{2,d}$ by $\psi$ to the cylinder is $g$.
All that said, we may again define the following function of two-points on the cylinder:
$$\sigma(p,q)=-2\eta(\psi(p),\psi(q))\tag{5}.$$
My question here is: is $\sigma(p,q)$ the geodesic distance on the Lorentzian cylinder? More generally, is it just a coincidence that we can get the geodesic distance function on $\mathbb{R}^{1,d-1}$ using (2), or is it a more general result valid for other sections of the lightcone in $\mathbb{R}^{2,d}$?