$r$ and $R$ difference with Schwarzschild metric
$\let\a=\alpha \let\b=\beta \let\th=\theta \def\ra{{(r^3+\a^3)}} \def\sa{{(r_1^3+\a^3)}}$ Let's begin to put some firm points.
The coordinate we today call Schwarzschild's $r$ isn't his $r$, but $R=\ra^{1/3}$.
He calls $R$ an "auxiliary quantity". E.g. consider the form he gives to Kepler's third law (second-to-last equation of his paper): $$n^2 = {\a \over 2\,\ra}$$ with $n=d\phi/dt$.
Schwarzschild writes metric as $$ds^2 = F\,dt^2 - (G + H\,r^2)\,dr^2 - G\,r^2 (d\th^2 + \sin^2\!\th\,d\phi^2) \tag1$$ with $F$, $G$, $H$ functions of $r$ (eq. 6).
From (1) we see that angular part of the metric is different from the one we are accustomed to. In particular, surfaces $r=\rm const.$ don't have area $4\pi r^2$, but $4\pi r^2 G$ and $G$ is not constant.
It's clear from these and other aspects of Schwarzschild's paper that he considers $t$, $r$, $\theta$, $\phi$ as true physical space-time coordinates. In particular, he constrains the "singularity" to $r=0$, i,e, out of spacetime. He states that clearly when he writes condition (13), having precisely that aim.
Of course both $r$ and $R$ are legitimate radial coordinates. They are however not physically equivalent. In mathematical terms, using $r$ the spacetime manifold exhibits a singularity only at border $r=0$. Using $R$ that singularity becomes the horizon ($R=\a$) and it makes sense to ask oneself if points with $R<\a$ are to be included in spacetime. As is well known, a thorough understanding of that issue would have taken about half a century.
OK, maybe this is history of physics. Or it's physics in its own right?
Edit
From comments and answers I've read I deem necessary to expand somewhat my answer, hoping it will help to solve several doubts and misconceptions.
(A short historical note. Einstein's paper exhibiting the final form of his equations is dated Nov 25th 1915. Schw. paper is dated January 13th 1916. Schw. died - by an autoimmune disease still incurable today - on May 11th 1916. He was 43.)
First, there is a strong difference between our present way of understanding spacetime in GR and the way of E.'s and S.'s times. I already remarked that one century ago there still weren't clear ideas about the meaning of spacetime coordinates. There still was a tendency to consider them endowed of some physical significance by themselves. In particular, although S. was well aware that spacetime in his solution is curved and space sections are curved too (not euclidean), he writes a formula for angular velocity in a circular orbit $$n^2 = {\a \over 2\,\ra}$$ and concludes
the angular velocity does not, as with Newton's law, grow without limit when the radius of the orbit gets smaller and smaller, but it approaches a determined limit $$n_0 = {1 \over \a \sqrt2}.$$ (For a point with the solar mass the limit frequency will be around $10^4$ per second).
(remember that S.'s $\a$ is what is known as "S. radius" and is proportional to the mass of central body).
On the contrary, present-day approach, grounded on sounder mathematical bases, is roughly the following.
- Spacetime is a semi-Riemannian manifold of dimension 4 and signature $+---$.
1.1 A (real) manifold is a set wherein real coordinates may be defined. This can be done in several ways. Coordinates are nothing but labels for the set's points.
1.2 It's not required that a set of coordinates be able to cover the whole manifold. More sets are allowed - it's only required that together they cover the whole manifold and smoothly overlap between them. Each set is named a card and the ensemble is named an atlas.
1.3 An easy instance. To define a sphere as a (2D) manifold a minimum of two cards are required. Among geographers a lot of cartographic projections are in use and a world's atlas is a good example of the general idea.
- A Riemannian manifold is a manifold where a metric is defined. Roughly, a formula giving the distance between (infinitesimally) near points.
2.1 More exactly, the distance squared $ds^2$. So the metric must be positive definite.
2.2 In a semi-Riemannian manifold an extension is allowed: $ds^2$ may also be negative. (A contradiction? Not quite. It's enough to relax the intuitive interpretation as a distance squared.) Minkowski's spacetime of SR is already an instance of that: there are spacelike intervals, timelike ones, and lightlike too.
2.3 The signature refers to how many independent displacements have metric of each sign. In GR there are two conventions in use: $+----$ means timelike has positive $ds^2$, spacelike negative. $-+++$ is the opposite. There is no real difference - it's only necessary to consistently adhere to one convention. Mixing them in a calculation leads to certain disaster.
- Leaving aside more sophisticated usages, the metric is the only way we have to give coordinates a physical meaning. It allows to compute the time a clock measures between events or the length of a space interval and so on.
3.1 Assume a coordinate is called $t$. It's the initial of "time" in English, of "temps" in French, of "tempo" in Italian, of "tiempo" in Spanish ... but not of "Zeit" in German or "czas" in Polish. So why should we assume that coordinate means time? It may (and usually will) be, but not always. Only looking at the metric can we get a safe answer.
Now let's come back to S. He makes use of two radial coordinates: $r$ and $R$. But there's no doubt about which he takes as the "physical" radius. His paper's title says
On the Gravitational Field of a Mass Point according to Einstein's Theory
A "mass point". It's obvious that he locates that mass at $r=0$, that $r$ can take all real positive values, that he'll require no singularity appears for positive $r$. At paper's end, as I noted above, he computes the revolution period of a planet as a function of radius and comments on a peculiar result; that period doesn't go to zero with $r$. On the contrary, it goes to a non-zero limit of about $0.1$ ms. He doesn't ask himself what's the meaning of $r$ (physical distance from the central mass?) nor what time would be that $0.1$ ms - which clock would measure it.
No wonder: GR had just been born and E. himself wasn't much clearer about such matter. But after a century and a lot of valuable work of eminent theoreticians we can and must have sounder ideas.
As to $R$, I repeat that S. calls it an "auxiliary" quantity. In modern terms we would consider it a radial coordinate as good as $r$ - metric can be written both in terms of $R$ (S.'s eq. (14), exactly the same universally denoted today as "S.'s metric") and in terms of his $r$. S. doesn't write the latter but you can see it here: $$ds^2 = \left[1 -\a\,\ra^{-1/3}\right] dt^2 - {r^4 \over \ra\,\left[\ra^{1/3} - \a\right]}\,dr^2 - \ra^{2/3} (d\th^2 + \sin^2\!\th\,d\phi^2).\tag2$$
You can use eq. (2) to answer e.g. the following question: "Once fixed $t$ and $r$ you're left with a 2D surface (a sphere). What's its area?" The answer isn't $4\pi r^2$, but the more complicated $4\pi\,\ra^{2/3}$. You could use $R$ instead and then (from S.'s metric) you'd find $4\pi R^2$, which is the same. Another question could be: "What's the radius of that spehere?" S. wouldn't have hesitated. His answer would have been $$\int_0^r\! {r_1^2 \over \sqrt{\sa \left[\sa^{1/3} - \a\right]}}\,dr_1.$$ (The integral looks intimidating, but it's easily solved through a substitution - can you see it?) Surely S. would have preferred to use his "auxiliary quantity" writing the required radius as $$\int_\a^R\!\sqrt{R_1 \over R_1 - \a}\,dR_1.$$ He wouldn't have worried about the lower $\a$ limit, which isn't the origin of $R$ coordinate. To him the real radial coordinate was $r$.
It's up to us to be worried: if $r$ and $R$ are on equal footings as radial coordinates, where is the space origin? At $r=0$ or at $R=0$? A mathematician's answer would be straight: if you use $r$ then that coordinate works for all $r>0$ - only at $r=0$ metric (2) is singular as the coefficient of $dr^2$ vanishes. Instead if you want to use $R$ with S.'s metric you must keep $R>\a$ since metric becomes singular at $R=\a$.
In both cases - the mathematician would continue - this doesn't mean that your manifold ends there. It could, or it could go on - it's your choice and it doesn't depend on the radial coordinate you initially assumed. It's true that $r$ suggests that $r=0$ is the end of your manifold whereas $R$ naturally leads to think that there is "something out there", but here mathematicians take a different view. They would tell us "what you're looking for is whether the manifold defined by the chart you have built allows for an extension or not. If it doesn't, this is the end of our argument. If it does, it's to you to decide if you want to give a physical meaning to the extension, or not."
Well, the answer is that the extension exists. How can we say that? Simply by finding other coordinates which are able to cover a region wider than the original one. This was actually done, in several ways, and opened physicists a novel world. A physical meaning to that world was given in 1939, when a paper by Oppenheimer and Snyder ["On Continued Gravitational Contraction"] (https://journals.aps.org/pr/abstract/10.1103/PhysRev.56.455) introduced the idea we today know as gravitational collapse.