What induces and what is the meaning of the conformal factor in the Poincaré ball? I'd like to understand why the conformal factor for the Poincaré ball usually is defined as follows:
$$
\lambda_{x}=\frac{2}{1+\kappa\|x\|_2^2}
$$
where $\kappa$ is the sectional curvature of the Poincaré ball.
I've come across the following document (Theorem 1.5, page 1), which seems to say that the conformal factor is uniquely defined, up to a scaling constant.
http://pi.math.cornell.edu/~bowman/metrics.pdf
I wonder why one usually uses a 2 in the numerator, instead of a 1. Is there some reason (e.g., conventions in physics) that there's the convention to use a 2 in the numerator?
The conformal factor plays an important role in the metric $g_x$. As the metric is defined as $g_x=\lambda_x^2 I$ where $I$ is the identity matrix.
There's a lot of literature that shows how Euclidean geometry is recovered when the curvature $\kappa\to 0$. However, with the current conformal factor, the Euclidean geometry would have the metric
$$
\lim_{\kappa\to 0} g_{x}=4I.
$$
It would be nicer to use a conformal factor with a 1 in the numerator to get exactly the Euclidean metric tensor $I$.
So I'd like to know what are the reasons behind choosing the conformal factor with a 2 in the numerator and not a 1. And I'd like to know what induces it.

What I've thought of so far:
Let's start with the minimal requirements for a Riemannian manifold. The smooth manifold $M$, and the Riemannian metric $g$. So we have the Riemannian manifold $(R,g)$. Now, actually, we're free to define $g$ as we'd like to, as long as it's a Riemannian metric. 
In the case of the Poincaré ball, the conformal factor in $g_x=(\lambda_x)^2 I$ just defines how a vector is interpreted in the tangent space at $x$ at a certain point.
For example, when we compute the tangent space norm, the same vector $v\in\mathbb{R}^d$ is interpreted differently at different tangent spaces. For example,


*

*The norm of $v$ is only scaled up by a factor of 2 at the origin (as the conformal factor has a 2 in the numerator and the denominator becomes 1)

*The same $v$ represents a much larger length if it's a vector at the tangent space of a point that is close to the border of the Poincaré ball. Then the denominator in the conformal factor becomes infinitely small and thus the norm infinitely large.


So, I can understand how the conformal factor accounts for the interpretation of the scale of tangent vectors. It seems that the important part, relevant for the behaviour of the scaling is only in the denumerator. Can one arbitrarily choose the value of the numerator? Why is it usually chosen to be 2?
It seems that changing the scale of the conformal factor is equivalent to changing the scale of the basis vectors of the tangent spaces at some point x. To me it seems that nothing should prohibit me from freely choosing the scale of these basis vectors. Still, why did people prefer to have a scale-up of 2 at the origin instead of 1? Where does the 2 come from?
My current guess is that it comes from the conventions in tensor calculus. I've spent some more thought on it and I realized the following: A point inside the Poincaré ball is some way of specifying some true geometric quantity. So, the Poincaré ball can be seen as a coordinate system. And its covariant basis stems from the derivating the true geometric quantity w.r.t. the coordinates on the Poincaré ball. Now, I wonder what the true geometric quantity is. It might be that it's just a point on the Hyperboloid in the Euclidean ambient space. So derivating the projection from the Poincaré ball to points on the hyperboloid in the Euclidean ambient space might be what results in this covariant basis that gives the conformal factor that scales the covariant metric tensor.
 A: If the metric is multiplied by some (constant) factor, then the curvature is divided by the same factor.
We can have either
$$ds^2=\left(\frac{2}{1-\lVert x\rVert^2}\right)^2\lVert dx\rVert^2,\quad\kappa=-1,\quad r=1,$$
or
$$ds^2=\left(\frac{1}{1-\lVert x\rVert^2}\right)^2\lVert dx\rVert^2,\quad\kappa=-4,\quad r=1,$$
or
$$ds^2=\left(\frac{1}{1-\tfrac14\lVert x\rVert^2}\right)^2\lVert dx\rVert^2,\quad\kappa=-1,\quad r=2$$
(where $r$ is the radius of the ball; $\lVert x\rVert<r$). There is always a factor of $2$ or $4$ somewhere; we can't get rid of them completely, unless we use a different coordinate system. By convention, $\kappa=-1$. And we're already familiar with unit spheres; have you ever heard of a bit sphere ($r=2$)? Neither have I. By the process of elimination,
$$ds^2=\left(\frac{2}{1-\lVert x\rVert^2}\right)^2\lVert dx\rVert^2.$$

I don't quite understand what you're trying to say in your last paragraph. Yes,  the Poincare ball can be seen as a coordinate system. And we should try to work with "true geometric quantities" (such as a hyperbolic polygon's side lengths) that don't depend on coordinate systems, so this factor of $2$ in the numerator becomes irrelevant.
