There are several definitions of "mean" in the Poincaré ball, an isometric hyperbolic space model. The Poincaré ball $\left( \mathbb{D}^n, \mathfrak{g}^{\mathbb{D}} \right)$ is defined as a manifold $$\mathbb{D}^n_{\kappa} = \left\{ \mathbf{x} \in \mathbb{R}^n : \left\Vert \mathbf{x} \right\Vert_2 < -\frac{1}{\kappa} \right\} ~~ (\kappa < 0)$$ with the Riemannian metric $$\mathfrak{g}^{\mathbb{D}_{\kappa}}_{\mathbf{x}} = (\lambda^{\kappa}_{\mathbf{x}})^2 \mathfrak{g}^{\mathbb{E}} $$ where $$\lambda^{\kappa}_{\mathbf{x}} := \frac{2}{1 + \kappa \left\Vert \mathbf{x} \right\Vert_2^2} $$ Here, $\kappa < 0$ is the curvature, and $\mathfrak{g}^{\mathbb{E}} = \mathbf{I}_n$ is the Euclidean metric tensor.
First type of mean is the Möbius gyrocentroid, which is transformed from Einstein gyrocentroid defined in Klein model:
Given a set of points $\left\{ \mathbf{x}_i \in \mathbb{D}_{\kappa}^n \right\}_{i=1}^{N}$ with positive weights $\left\{ w_i \in \mathbb{R}_+ \right\}_{i=1}^{N}$, the Möbius gyrocentroid of them is defined as $$\bar{\mathbf{x}} = \frac{1}{2} \otimes_{\kappa} \left( \frac{\sum_{i=1}^{N} w_i \lambda_{\mathbf{x}_i}^{\kappa} \mathbf{x}_i}{\sum_{i=1}^{N} w_i (\lambda_{\mathbf{x}_i}^{\kappa} - 1) } \right) $$
Here, "$\otimes_{\kappa}$" is the Möbius scalar-vector multiplication: $$r \otimes_{\kappa} \mathbf{x} = \frac{1}{\sqrt{|\kappa|}}\tanh \left( r \tanh^{-1} \left( \sqrt{|\kappa|} \left\Vert \mathbf{x} \right\Vert_2 \right) \right) \frac{\mathbf{x}}{\left\Vert \mathbf{x} \right\Vert_2} $$
Another kind of mean if the Fréchet mean, which is defined as the point that minimizes the Fréchet variance, i.e.
Given a set of points $\left\{ \mathbf{x}_i \in \mathbb{D}_{\kappa}^n \right\}_{i=1}^{N}$ with positive weights $\left\{ w_i \in \mathbb{R}_+ \right\}_{i=1}^{N}$, the Fréchet mean of them is defined as the solution of the following optimization problem: $$\bar{\mathbf{x}} = \arg\min_{\mathbf{z} \in \mathbb{D}_{\kappa}^n} \frac{1}{N} \sum_{i=1}^N w_i \cdot d_{\mathbb{D}}^{\kappa}(\mathbf{z}, \mathbf{x}_i)^2 $$
Here, $d_{\mathbb{D}}^{\kappa}(\cdot, \cdot)$ refers to the geodesic distance in Poincaré ball, which can be computed as $$ d_{\mathbb{D}}^{\kappa} (\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{|\kappa|}}\cosh^{-1} \left( 1- \frac{2 \kappa \left\Vert \mathbf{x} - \mathbf{y} \right\Vert_2^2}{(1 + \kappa \left\Vert \mathbf{x} \right\Vert_2^2)(1 + \kappa \left\Vert \mathbf{y} \right\Vert_2^2)} \right) $$
I am confused about the differences between above two means. Are they the same in the Poincaré ball whenever the values of weights are ($w_i \equiv 1$ for all $i$s or $w_i$ can be arbitrary positive real number)? What are the purposes of designing two types of means?