Conserved quantities in GR
In GR, energy (or mass) is typically an ill-defined concept. In flat spacetime, we define energy as the conserved quantity corresponding to time translational symmetry. Extending this to GR is quite tricky mainly because, what one is calling time is already observer dependent (this is of course also true in flat spacetime, but at least there we have a canonical definition of time given by inertial observers). A second problem in GR is that time translation may not be a symmetry of the space-time, making it impossible to define energy. In particular, recall that the metric in GR is a fluctuating field, which makes it doubly hard to define timelike Killing vectors when the background itself is fluctuating.
Anyway, I hope what you can get from this is that defining energy and in fact any conserved quantity that depends on isometries of the space-time is not really something one can talk about in general relativity. So what do we do? How do we define such quantities?
How to define energy in GR?
One possible solution is to go very very far away from all forms of matter in a region where only radiation may exist. In this region - known as asymptotic infinity - spacetime is approximately flat, and one may hope to define energy here. In this region, we have a well defined notion of inertial observers w.r.t. whom we may define time and hence energy. The energy/mass so defined is called the ADM (Arnowitt, Deser, Misner) energy of the space-time. It describes the mass of the system as measured by an inertial observer sitting at infinity.
ADM mass of the Schwarzschild Black Hole
The precise formulae for the ADM mass can be read off for instance in Carroll. Using that formula, we can compute the ADM mass of the Schwarzschild black hole and we find that it is $M$. This is how we know that the quantity $M$ represents the mass of the Schwarzschild Black Hole. In other words, the statement is, place an inertial observer very far away from the black hole and ask him/her to measure the energy of the system which he/she will do w.r.t. the time that he/she is experiencing. The result they will find is that the energy of the system $=M$.
A caveat here is that they must make sure that they are themselves at rest w.r.t. the black hole. There is a wide class of inertial observers at infinity, some (actually, most) of which are moving relative to the black hole. We would like to define mass as the energy of the system at rest. Thus, we must choose our inertial observer so that the momentum that he/she measures is zero. In this frame, the energy that he/she measures will be the mass. When this is done for Schwarzschild, the answer we get is $M$.
A side note
The ADM mass is what we would typically like to call mass of a system, except that it lacks in one respect. An inertial observer at infinity, is not able to measure energy in gravitational or electromagnetic radiation that is emitted. For instance, if the Schwarzschild black hole were to start radiating energy via gravitational waves and eventually disappear, the ADM mass measured by the observer at infinity would still be $M$.
When gravitational radiation is important (for instance, when studying scattering of gravitational waves) for the problem, a more convenient definition of the mass is the Bondi mass $m_B$ which is defined as the mass measured by a Bondi observer at infinity. A Bondi observer is one that moves at the speed of light along null infinity. The Bondi mass is a function of (null) time $m_B(u)$ so that it captures not only the current mass, but also the change in the mass of the system due to radiation.
The causal structure, defined by light cones can be shown in t-r plane. The slope of the cones given by
\begin{equation}\tag{1}
\frac{dt}{dr} = \pm \frac{1}{(1-\frac{r_S}{r})}
\end{equation}
increases to infinity for $r\rightarrow r_S$. (first picture below) Hence light rays asymptotically 'reaches' Schwarzschild radius in this coordinate system. The idea of tortoise coordinate is to make $\frac{dt}{dr}$ smaller. Just by integrating (1), we get $r^* = r+r_S \text{ln}(\frac{r}{r_S}-1)+\text{const}$. We can now map $r < r_S$ using tortoise coordinates ($t,r^*$) in which Schwarzschild metric beomes,
\begin{equation}
ds^2 = -(1-\frac{r_S}{r})(dt^2 + dr^{*2}) + r^2d\Omega^2)
\end{equation}
(because $dr^* = dr/(1-\frac{r_S}{r}))$
Now $dt/dr^*$ is a constant hence we have light cones which are not asymptotic in $t-r^*$ plane (second picture).
The proper time and coordinate time can be related (from geodesic equation) as,
\begin{equation}\tag{2}
\frac{d\tau}{dt} = (1-\frac{r_S}{r})^{1/2}
\end{equation}
Hence, distant observer (named B) will observe light coming from infalling observer (name him A), red shifted by (one over) this factor (and also we cannot define once $r<r_S$ - physically A appears not only to be still but gets reddened and hence eventually dimmer to B).
A reaches $r_S$ in finite proper time but for B at rest, this would take infinite time. In other words, $r_S$ forms the Cauchy Horizon beyond which we have unique geodesics but within it, is a singularity (which does not belong to the Lorentzian manifold) with no future null-like geodesics.
So from (2) it is clear that, even though, A crosses $r_S$, B cannot see this event. Even though A can hold meeting inside the Horizon, B will never know of it. Is there a transformation $t\rightarrow\tau$? All such transformations are affine, in other words, proper time (atleast for null-like geodesics) are affine parameters which appear in the geodesic equations. But from the very definition of singularity, future null geodesics do not exists.
Best Answer
I am not sure that I properly understand the question. However, looking at the paper in the link you see that the relevant equations are two $$\frac{dt}{d\tau}=e(1 - 2GM/r)^{-1}$$ and $$\frac{dr}{d\tau} = \pm \sqrt{e^2 -\left(1 + \frac{C}{r}\right)\left(1 + \frac{\ell^2}{r^2}\right)}$$ where $e$ and $\ell$ are constants.
In the paper, an issue is made explicit:
"Because the time coordinate behaves differently in this case, is it still correct to identify it with the proper time of an observer at infinity? "
Yes it is: If a particle stays at rest at infinity then both $r\to +\infty$ and $\frac{dr}{d\tau}\to 0$ there. That is possible only if $e=0$ from the second equation. The first equation implies that $t= \tau + constant$. Hence, again $t$ is the proper time of a particle at rest very far from the singularity.
Regarding the fact that negative mass "speeds up" (proper) time, I think that the meaning of that statement is as follows. Look at the plots for $x>1$. The red line is $\frac{d\tau}{dt}$ for positive mass, the blue one is the analog for negative mass. Here Schwarzschild radius is at $x=1$ as $2GM =1$. The two plots can be actually compared only in the "positive-mass external" region $x>1$. You see that the "speed of the proper time vs (Killing) time" $\frac{d\tau}{dt}$ increases while approaching $x=1$ and diverges around the singularity $x=0$ along the blue line (negative mass). Conversely, it decreases along the red line (positive mass).
However, far from away the singular region, i.e., for $x\to +\infty$ the two curves coincide.