Let me recall some basic computational facts (the meaning of which I am trying to understand).
In flat space, an adapted coordinate $(\tau, \xi)$ associated with an observer with constant proper acceleration $\alpha$ is $$ ct = \frac{c^2}{\alpha} e^{\alpha \xi/c^2} \sinh \big( \frac{\alpha}{c}\tau \big), \quad x = \frac{c^2}{\alpha} e^{\alpha \xi/c^2}\sinh \big( \frac{\alpha}{c}\tau \big) \ . $$ Quantizing the free real scalar theory in $(t, x)$ and $(\tau, \xi)$ leads to an Unruh temperature $$ T_\text{Unruh} = \frac{\hbar \alpha}{2\pi c k_B} \ . $$
The Schwarzschild coordinate $(t,r, \theta, \varphi)$ can be transformed to a $(t, \xi, \theta, \varphi)$ coordinate, such that $$ r = r_s + \xi \ . $$ Near the horizon, $\xi/r_s$ is small, and the metric takes the form $$ ds^2 = - \frac{\xi}{r_s}c^2 dt^2 + \frac{r_s}{\xi}d\xi^2 + ds^2_{\theta, \varphi} + \text{higher $\xi$ terms} \ . $$ One can further transform it to a "flat coordinate" $$ cT = 2r_s e^{\frac{\xi}{2r_s}} \sinh \frac{c}{2r_s}t, \qquad X = 2r_s e^{\frac{\xi}{2r_s}} \cosh \frac{c}{2r_s}t \ . $$ The near horizon metric now reads $ds^2 = -c^2 dT^2 + dX^2 + ds^2_{\theta, \varphi}$.
Now, if I "compare" the $(t, x) \to (\tau, \xi)$ transformation with the $(T, X) \to (t, \xi)$ transformation in the two bullet points, I would say that there is an "effective proper acceleration" $$ \frac{c}{2r_s} = \frac{\alpha_\text{BH}}{c}, \qquad \alpha_\text{BH} = \frac{c^2}{2r_s} = \frac{c^4}{4GM} = \text{finite surface gravity} \ . $$ Translating this acceleration to a "Unruh temperature", one recovers the standard formula $$ T = \frac{\hbar \alpha_\text{BH}}{2\pi c k} = \frac{\hbar c^3}{8\pi G k M} \ . $$
Here are my confusions
Does the $\alpha_\text{BH}$ have the meaning of "proper acceleration" of something measured by some actual local comoving observer near the horizon?
The above computation is performed in the near horizon limit, leading to the standard Hawking temperature. However, accrding to this answer, this is the temperature measured by a faraway observer. I wonder how to understand this "near horizon" and "faraway" relation? How do we know if some computation performed at some location (e.g. near horizon) = something measured by a faraway observer?
